‘ “RWY Michigan 5% Universfiy . _..._—-—-————-— This is to certify that the thesis entitled Studies of Plain and Reinforced Frozen Soil Structures presented by Sweanum 800 has been accepted towards fulfillment of the requirements for Ph.D. Civil Engineering degree in Major professor [INe Feb. 23, 1984 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution STUDIES OF PLAIN AND REINFORCED FROZEN SOIL STRUCTURES By Sweanum $00 A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Sanitary Engineering 1983 -.-v|U .H. u e r o v :5 .x. r A .1 P .—. n u 9 .. a . o r . . o r . . ... .CU . u u u d . n It I .u» .v. . . .4.- I il b 31 o v A v . O '1 t u . a . A an» O u . u » .tn ABSTRACT STUDIES OF PLAIN AND REINFORCED FROZEN SOIL STRUCTURES By Sweanum Soo Plain and reinforced frozen sand beams with cross sections 3.25 inches by 2.75 inches were loaded incrementally in pure bending at -6°C and ~10°C. For plain beams, the deflection rate (during secondary creep) showed a linear relationship with applied load on a log-log scale while experimental results of beams reinforced with a single 3/16-inch diameter steel bar showed a bi-linear relation- ship. The reinforced beams at low loads deformed at a slower rate than that for the plain beams. After a ”critical load" was reached, the rate increased to approxi- mately that of the plain frozen beams. At the critical load. the incremental shear force transfer between the rein- forcement and the soil dropped drastically. For purposes ofanalysis, a plain beam was divided into a number of layers through its depth and a number of seg- ments along its length. Uniaxial elements and beam theory were used. An incremental approach was applied to calculate the creep deformation in the time domain. The power creep law and the hyperbolic sine creep law were employed with distinct creep parameters in the tension and compression layers. Numerical results showed good agreement with experimental data. The results indicated that the tension 4’. ;v. i | r. \ V . A O n S I r\: .v- hu- r a a c on. A .II II. 'r I ... . r .u «1.14 a r a P u r r u u .. i P . Au. 9.3 ~\\' Q U 1" C O O P r r r. L. t h I!‘ II. . I UH. Sweanum 800 parameters had a much greater effect on the beam behavior than the compression ones. Using a single set of average values leads to erroneous results. Finite element models involving multiaxial stress states in frozen soil were also developed. The difficulty arising from differences in material properties for tension and compression was circumvented by postulating the exis- tence of an "effective creep strain rate" and an "effective stress." A weighting procedure, based on the three prevail- ing principal stresses, was proposed to evaluate these "effective" parameters. For reinforced frozen soil members, the bond behavior between the reinforcement and soil was modeled by certain bond interface elements with nonlinear properties inferred from experiments. An analysis of a reinforced frozen sand beam under a constant load showed that the stress in the frozen sand con- tinued its transfer to the reinforcement as time elapsed until a steady-state condition was reached. An analysis of an axisymmetric system with an embedded reinforcing bar being "pulled out" of a frozen soil mass showed that the bond stress distribution initially was nonlinear but would gradually become uniform with time. To My Parents, Lee-Shuang and Kuo-Shih 500 ii HL 4 J .‘d P .4.“ an» I D A ‘ -\- n i V .4. h-u A i l n I P a II a I. .u (a . ta . A ‘44 A. in ‘ .\.U .. . ..II o n r I v on. o u . a v 1 P... a .- ACKNOWLEDGMENTS The writer wishes to express his appreciation to his major professors, Dr. Robert K. Wen and Dr. Orlando 8. Andersland, Professors of Civil Engineering, for their guidance and numerous helpful suggestions during the con- ducting of the research and preparation of this dissertation. Thanks also to members of the writer's doctoral committee, Dr. Nicholas J. Altiero and Dr. Gary L. Cloud, Professors of Metallurgy, Mechanics and Materials Science. Gratitude is extended to the Department of Civil and Sanitary Engineering and the National Science Foundation for supporting this work. The writer also owes his appreciation to his parents, Lee-Shuang and Kuo-Shih $00, for their financial support for the writer to pursue his PhD degree. Special thanks is also due his wife, Tzongyun Katy Wu, and his host family, Jerry and Valerie Nilson, for their encour~ aQement and spiritual support. LIST OF LIST OF LIST OF CHAPTER 1.1. 1.2. 1.3. CHAPTER 2.1. 2.2. 2.3. TABLE OF CONTENTS TABLES ......................................... vii FIGURES ........................................ ix SYMBOLS ........................................ xiv I. INTRODUCTION ............................. 1 General ........................................ 1 Objective and Scope ............................ 1 Review of Literature ............................ 3 1.3.1. Mechanical Properties of Frozen Soils 3 1.3.2. An Engineering Theory for Frozen Soils 11 1.3.3. Structural Members in Frozen Soils ... 17 1.3.4. Flexural Creep Behavior of Structures in the Uniaxial Stress State ......... 23 1.3.5. Analysis of Structural Creep Behavior in a Multi-axial Stress State ........ 25 1.3.6. Interface Models for Finite Element Analysis ............................. 30 II. MODEL BEAM TESTS 0N FROZEN SOIL .......... 50 General ........................................ 50 Materials and Sample Preparation ............... 50 EClUipment and Test Procedures .................. 52 2.3.1. Equipment ............................ 52 2.3.2. Test Procedures ...................... 54 iv r o . A v nP.‘ .u» r1 rt PAH vu rh Pb |H . I 1" I C I .1. 7s . putv rhv uci «(n .11. Aid. CHAPTER wwwwww boom CHAPTER Experimental Results and Discussion ............ 56 2.4.1. Experimental Results ................. 56 2.4.2. Discussion ........................... 59 III. ANALYSIS OF FROZEN SOIL BEAM USING ONE- DIMENSIONAL FINITE ELEMENTS .............. 92 General ........................................ 92 Method of Analysis ............................. 93 Creep Laws ..................................... 95 Creep Parameters ............................... 98 Computer Program ............................... 100 Numerical Results and Discussion ............... 102 3.6.1. Analysis of a Simply Supported Beam by the Power Creep Law .................. 103 3.6.2. Analysis of a Simply Supported Beam by the Hyperbolic Sine Creep Law ......... 105 3.6.3. Parametric Studies .................... 106 IV. ANALYSIS OF FROZEN SOIL STRUCTURES USING THO-DIMENSIONAL FINITE ELEMENTS .......... 144 General ........................................ 144 Finite Element Formulation for Two-Dimensional Frozen Soil Elements ........................... 145 Computation of Multi-Axial Creep Strains in Frozen Soil .................................... 149 Material Model for the Bond Interface .......... 155 Computer Program ............................... 160 Numerical Results and Discussion ............... 163 4.6.1. Analysis of a Plain Frozen Sand Beam . 163 V l""\ \1 ..., |-x..‘q' n-,. 5 .1 ‘ .v ‘1 5v - I. o \ \..- ‘1'. V‘ - A a n IA;- 1 u... v- ... A, fi__ ‘ b .5- "' v I‘.~ X 7‘ V .2. Analysis of a Reinforced Frozen Sand Beam ................................. 166 4.6.3. Analysis of the ”Pull Out" Test with a Constant Load ........................ 168 4.6.4. Analysis of the "Pull Out" Test with a Prescribed Displacement Rate ......... 169 CHAPTER V. SUMMARY AND CONCLUSIONS .................. 195 5.1. Summary ........................................ 195 5.2. Concluding Remarks ............................. 197 LIST OF REFERENCES ..................................... 201 APPENDIX A. MODEL BEAM TEST DATA ...................... 207 APPENDIX B. COMPUTER PROGRAM FOR ONE-DIMENSIONAL FINITE APPENDIX C. ELEMENT ANALYSIS .......................... 223 COMPUTER PROGRAM FOR TWO-DIMENSIONAL FINITE ELEMENT ANALYSIS .......................... 239 vi LIST OF TABLES Summary of Long-Term Cohesion for Frozen Soils ........................................... 33 Summary of m Coefficient ........................ 34 n Value for Different Types of Piles at -6°C .... 34 Summary of Test Results for Sample 3 (Plain Frozen Sand Beam at About -10°C) ......... 64 Summary of Test Results for Sample 4 (Plain Frozen Sand Beam at About -6°C) .......... 64 Summary of Test Results for Sample 5 (Frozen Sand Beam Reinforced with a 3/16-inch Smooth Steel Rod at about -10°C) ................ 65 Summary of Test Results for Sample 6 (Frozen Sand Beam with a 3/16-inch Steel Rod. with a 1/16-inch Lug at Each End at about -10°C). 65 Summary of Initial Deflections at the Beginning of Each Load Increment for Sample 3 (Plain Frozen Sand Beam, -10°C) ................. 66 Summary of Initial Deflections at the Beginning of Each Load Increment for Sample 4 (Plain Frozen Sand Beam, -6°C) ................................ 67 Estimates of Initial Tangent Moduli for the Frozen Sand Beams at -10°C ...................... 68 Estimates of Initial Tangent Moduli for the Frozen Sand Beams at -6°C ....................... 69 Comparison of Materials Used in Model Beam Tests with Those Used by Bragg (1980) and Eckardt(1981) 110 Creep Parameters Obtained by Eckardt (1981) ..... 111 Values of Parameters for Power Creep Law ........ 111 vii .4. .5. Creep Parameters Obtained by Bragg (1980) for the Power Creep Law ............................. 112 Values of Parameters for the Hyperbolic Sine Creep Law (Extrapolated from Bragg's Data) ...... 112 Estimate of the Initial Bond Modulus from Experi- mental Results of the "Pull Out" Tests Conducted by Alwahhab (1983) .............................. 171 viii “r “V II‘ “v Figure LIST OF FIGURES (a) Open Excavation Supported by Straight Frozen Walls; (b) Open Excavation Supported by a Curved Wall; (c) Tunnels Supported by Frozen Malls ...... 35 Volume Concentration of Ottawa Sand and Peak Strength ......................................... 36 Temperature Dependence of Unconfined Compressive Strengths for Several Frozen Soils and Ice ....... 37 Constant-Stress Creep Test: (a) Basic Creep Curve; (b) True Strain Rate vs. Time ............. 38 Typical Stress-Strain Curves for Unconfined Compression Tests ................................ 39 Comparison of Compression Strain Rates Reported by Different Authors ................................ 40 Schematic Representation of the Whole Failure Envelope for Frozen Ottawa Sand .................. 41 Time Dependent Failure Envelopes ................. 42 Straight-Line Approximation of Time Dependent Failure Envelopes ................................ 43 Load-Displacement Curves for "Pull Out" Tests at 4 Different Temperatures ........................... 4 Load-Displacement Curves for "Pu1100ut" Tests at 45 Different Displacement Rates at -6 C ............. Load-Displacement Curves for "Pull°0ut" Tests at 46 Different Displacement Rates at -2 C ............. (a) Pure Bending of a Bisymmetric Beam; (b) Stress Distributions in Bending of a Rectangular Beam 47 for Various Values of the Power Parameter n ...... Bond-Link Element Developed by N90 and Scordelis 48 (1967) ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Ah ~04. pc. or .H» ‘4‘ D\ U »\Iv pin,- .15. a. n v .P. f u V Cr: PF! .11.. .. a .m. .J¢ “Irv (U TI; A). ... II.‘ a: r- g V F . Alli 7L. .15. .10. .11. .13. .14. .15. (a) The Joint Element Developed by Ghaboussi, et. al (1973); (b) The Joint Element in Local Coordinates ..................................... Diagram of Test System: (a) (Front View); (b) (Side View) ................................. Loading and Support System ...................... (a) Load, Support and Deflection Transducer Positions; (b) Deflection Transducer Positions in Plan View ....................................... Deflection at Mid-Span vs. Time Curves for Sample 3 (Plain Frozen Sand Beam, -10°C) ........ Deflection at Mid-Span vs. Time Curves for Sample 4 (Plain Frozen Sand Beam, ~6°C) ......... Deflection Rate at Mid-Span vs. Load for Plain Frozen Sand Beams (Log-Log Plot) ................ Deflection at Mid-Span vs. Time Curves for Sample 5 (Reinforced Frozen Sand Beam, -10°C) Deflection at Mid-Span vs. Time Curves for Sample 6 (Reinforced Frozen Sand Beam with Lugs at Both Ends, -10°C) ....................... Deflection Rate at Mid-Span vs. Load on Reinforced Frozen Sand Beams at -10°C (Log-Log Plot) ....... Comparison of Deflection Rate for Plain and Reinforced Frozen Sand Beams at -10°C (Log-Log Plot) ........................................... Sample 3 (Plain Frozen Sand Beam) After Deformation ..................................... Sample 5 (Frozen Sand Beam Reinforced with a Smooth Bar) Failed Due to Bond Slip ............. Bond Slip Within Sample 5 (Frozen Sand Beam Reinforced with a Smooth Bar) ................... Rthure of Sample 6 (Frozen Sand Beam Reinforced with a Steel Bar with Lugs) ...................... Derivation of Equation 2.4 ...................... Discrete System of a Beam ....................... 49 71 72 73 74 77 80 81 85 88 91 113 3.13. 3.14, (a) Strain in a Typical Layer; (b) Relation of Initial, Creep and Total Strain in a Time Interval ....................................... 114 Stress as a Function of Strain Rate for the Hyperbolic Sine Creep Law ....................... 115 Compressive Test Results Obtained by Bragg (1980) Plotted on a Semi-Logarithmic Scale ............. 116 Tensile Test Results Obtained by Bragg (1980) Plotted on a Semi-Logarithmic Scale ............. 117 Comparison of the Hyperbolic Sine Creep Law Curve with Uniaxial Creep Test Results ................ 118 Comparison of Numerical Results Based on the Power Creep Law with Experimental Results (Plain Frozen Sand Beam, -10°C) ........................ 119 Comparison of Numerical Results Based on the Power Creep Law with Experimental Results (Plain Frozen Sand Beam, -6°C) ......................... 123 Comparison of Numerical Results Based on the Power Creep Law with Experimental Results at Steady State Creep Deflection Rate (Log-Log Plot) 126 Stress and Strain Distribution on a Cross-Section at Different Time ............................... 127 (a) Stress Path of Time Hardening Theory; (b) Stress Path of Strain Hardening Theory ...... 128 Comparison of Numerical Results Based on the Hyperbolic Sine Creep Law with Experimental Results (Plain Frozen Sand Beam, -10°C) ......... 129 Comparison of Numerical Results Based on the Hyperbolic Sine Creep Law with Experimental Results at Steady State Creep Delfection Rate (Log-Log Plot) .................................. 133 Comparison of Numerical Results Based on the Hyperbolic Sine Creep Law and the Power Creep Law with Data from Bragg (1980) ..................... 134 Comparison of Numerical Results for Different Methods of Solution ............................. 135 Comparison of Numerical Results for Different Hardening Theories of the Power Creep Law ....... 136 xi no an» F» a «V. F»; .17. .18. .19. .20. .21. .22. .23. Comparison of Numerical Results Based on the Power Creep Law with Different Values of Moduli Ratio ........................................... 137 Comparison of Numerical Results Based on Hyper- bolic Sine Creep Law with Different Values of Moduli Ratio .................................... 138 Comparison of Numerical Results Based on the Power Creep Law with Creep Parameters from Eckardt (1981) and Varying Parameter 0C ......... 139 Comparison of Numerical Results Based on the Power Creep Law with Creep Parameters.from Eckardt (1981) and Varying Parameter cc ......... 140 Comparison of Numerical Results Based on the Power Creep Law with Creep Parameters from Eckardt (1981) and Varying Parameter n .......... 141 Comparison of Numerical Results Based on the Power Creep Law with Creep Parameters from Eckardt (1981) and Varying Parameter b .......... 142 Comparison of Numerical Results Based on the Power Creep Law with Average Creep Parameters of Tension and Compression with Experimental Results 143 A Quadrilateral Isoparametric Element ........... 172 Constitutive Relationships of the Bond Interface Element in the Direction: (a) Normal to the Bond Interface; (b) Parallel to the Bond Interface ... 173 (a) Relative Displacements Parallel to the Bond Interface; (b) Relative Displacements Normal to the Bond Interface .............................. 174 The Bond Interface Element in: (a) Global 5 Coordinates; (b) Local Coordinates .............. 17 Grid for the Analysis of a Plain Frozen Soil Beam 176 Comparison of One- and Two-Dimensional Finite Element Solution with Experimental Results for a 177 Plain Frozen Soil Beam .......................... Grid for the Analysis of a Reinforced Frozen Soil 17B Beam ............................................ Creep Parameters Measured from the "Pull Out" Test Results Obtained by Alwahhab (1983) ........ 179 xii 4.9. 4.10 4-11 4-12 4- 13. 4-14. 4 -‘15. 4-16. 4-17. 4- 18. 4-19. 4-20. 4.21. 4.22. 4.23. Comparison of Two-Dimensional Finite Element Solution with Experimental Result for a Reinforced Frozen Soil Beam ...................... 180 Bond Stress Distribution Along the Reinforced Layer: (a) On Top of the Reinforced Layer; (b) At the Bottom of the Reinforced Layer ........ 181 (a) Tensile Stress in the Reinforced Layer; (b) Total Tensile Force and Bond Force in the Reinforced Layer at t=10 hours ................... 182 Stress Distribution (in the X-Direction) at the Cross Section Near the Mid-Span of the Beam ...... 183 Grid for the Analysis of the "Pull Out" Test ..... 184 Comparison of Two-Dimensional Finite Element Solution with Experimental Result for the "Pull Out" Test with a Constant Load ............. 185 Stress Distribution Along the Steel Bar for the "Pull Out" Test with a Constant Load .............. 186 Stress Distribution in Frozen Soil at T=0 for the "Pull Out" Test with a Constant Load .......... 187 Stress Distribution in Frozen Soil at T=10 Hours for the "Pull Out” Test with a Constant Load ...... 188 Stress Distribution in Frozen Soil at T=3O Hours for the "Pull Out" Test with a Constant Load ..... 189 Comparison of Numerical Result with Experimental Result for the "Pull Out" Test with a Prescribed Displacement Rate ................................ 190 Stress Distribution Along the Steel Bar for the "Pull Out" Test with a Prescribed Displacement Rate ............................................. 191 Stress Distribution in Frozen Soil at T=0 for the "Pull Out" with a Prescribed Displacement Rate ... 192 Stress Distribution in Frozen Soil at T=5 Minutes for the "Pull Out" Test with a Prescribed Displacement Rate ................................ 193 Stress Distribution in Frozen Soil at T=20 Minutes for the "Pull Out" Test with a Prescribed Displacement Rate ................................ 194 LIST OF SYMBOLS /\ = area of cross section; B = beam width; [E3] = strain-displacement transformation matrix; 6T1"6S = relative displacements along n,x axes, respectively; 5 = relative displacement rate on the bond interface of two materials; 6: = relative creep displacement on the bond interface of two materials; 5(: "°sc’n’b = creep parameters for the creep law of the bond interface element; 5 i‘j = Kronecker delta; E = elastic modulus or initial tangent modulus; E53 ,Et,EC = respectively, effective, tensile and com- pressive elastic modulus; e = strain; 21,8C = respectively, instantaneous and creep strain; Eie iP = respectively, instantaneous elastic and plastic strain; effective creep strain rate; creep strain rate tensor; respectively, proof strain and proof stress for plastic deformation; creep parameters for power creep law; effective creep parameters for power creep law; compressive creep parameters for power creep law; tensile creep parameters for power creep law; stress; effective stress; effective strength; stress tensor; shear modulus; beam depth; second deviatoric stress invariant; bulk modulus; stiffness matrix of bond interface element; stiffness matrix of frozen soil element; structural stiffness matrix; bending moment; the relative displacement-d1splacement transformation matrix of the bond interface element; (n.S) ( 7‘ ,z,e) ’ (x,y) ( Xi,y) ( 5,11) i l L I l—ALW—J local coordiantes for bond interface element; pseudo-force vector of bond interface element; pseudo-force vector of frozen soil element; nodal force vector; structural pseudo-force vector; stress deviatoric tensor; shear strength; long-term shear strength; friction angle; long-term friction angle; interporation function; cylinder coordinates; strain energy; displacements along x,y axes, respectively; local coordinates for two-dimensional finite elements; structural global coordinates; local coordinates for isoparametric elements; incremental operator; column vector; row vector; rectangular matrix- 7.1. Eel The U'Ced fr Willie-r Kiel fr: 3I€3irej 33:1. .‘:n fr prg.' :53. CHAPTER 1 INTRODUCTION 1.1. General The purpose of this thesis was to study plain and rein- forced frozen soil structures. To achieve this, one- and two-dimensional finite element models were developed, and model frozen soil beams were tested. Computer programs were prepared to carry out the computation, and several examples of frozen soil structures were analyzed. Comparisons of numerical results with experimental results are presented. 1-2. Objective and Scope Frozen soil structures have been used to provide tempo- rary support for excavations, tunnels, and mine shafts for over one hundred years. In some cases, engineers were reluctant to use this type of structure because their load caDaCity could not be determined satisfactorily. Informa- tion on creep behavior and allowable loads are required to design frozen soil structures economically and safely. Previous studies on mechanical properties have shown that frozen soil is strong in compression and weak in ten- sion. Although there has been some work done on analysis Of frozen soil structures, they have not considered the 1 'l 'L (I! k, A, -) Cerium: '55 6713112 rilneerm difference of material properties in tension and compres- sion. Neglecting this difference may result in major errors in the analysis. A major objective of this study was to develop a method of analysis which includes the difference of material properties in tension and compression. Figure 1.1(a) shows examples of open excavations sup- ported by straight frozen walls. The mechanical behavior of these structures involves some beam actions. An investiga- tion of the flexural behavior of frozen soil, therefore, is desired. This investigation was first done experimentally by conducting model frozen soil beam tests. Then, the beam was analyzed using one-dimensional finite elements and engineering beam theory. Different material properties in tension and compression were used. Numerical solutions were carried out by a computer program and compared with experi- mental data. Figures 1.1(b) and (c) show examples of open excavation supported by a curved frozen wall and tunnels supported by frozen walls. These structure types may be considered as two-dimensional problems in axisymmetric and plane strain conditions. Two-dimensional finite elements for frozen soil were also developed in this work. The difficulty arising from the difference in material properties in tension and compression was circumvented by postulating the existence of an "effective creep strain rate" and an "effective stress. A general computer program for frozen soil structures in the plane stress. plane strain, and axisymmetric states was Ill written. Numerical solutions of frozen soil beams as plane stress problems were obtained and compared with numerical results of one-dimensional analysis as well as experimental data. Since frozen soil is strong in compression and weak in teansion, it appeared desirable to add reinforcement in the ‘teension zone to reduce beam deflections. To investigate the t3eehavior of such systems, reinforced frozen soil beams were £31.50 tested. Bond interface elements with zero thickness were used to model bond behavior between the reinforcement iBlWCi frozen soil. Combining bond interface elements and two- t3tained and compared with experimental results for the Y‘Eainforced frozen soil beams. Certain "pull out" tests were ii 150 analyzed to compare with the experimental data obtained by Alwahhab (1983). -3. Review of Literature -3.1. Mechanical Properties of Frozen Soils when ground temperatures drop below 0°C, the pore water Viithin the soil mass is frozen. Ice formed in the soil increases the bond between soil particles and the frozen soil strength can be compared to weak concrete. Due to the complex nature of ice and soils, the mechanical properties of frozen soil subjected to a loading are quite complicated. Many factors affect the mechanical properties of frozen frozen soils. Some of the previous studies are reviewed in the following sections. Effect of Particle Concentration: An important ctiaracteristic of frozen soil involves the relative propor- t ions of ice and mineral volume fractions. According to (3c>ughnour and Andersland (1968), the shear strength of a saturated frozen sand-ice mixture has a bilinear relation vvi th its sand volume fraction (Figure 1.2). At a low sand <2<3r1centration (ice-rich soil) the shear strength increased 3 l<3w1y with increased sand concentration. According to PiCJOKe, et a1. (1972), the increase in shear strength for low m ineral fractions was due to the increased deformation rate, which actually has a weakening effect on ice. 0n reaching a ‘3 t‘itical sand volume concentration. about 42%, friction t>etween sand particles and dilatancy begin to contribute and cHiiuse a rapid increase in shear strength (Goughnour and Andersland, 1968). Similar effects were shown in test results reported by I‘Céiplar (1971), where the critical sand volume fraction was ‘Found to be about 40%. More recently, Baker (1979) observed for an unsaturated frozen soil that shear strength drops rapidly with a reduction in ice content. Sayles and Carbee (1980) have found that a critical particle concentration close to 50% exists for a silt-ice When the silt concentration was less than 50%, the mixture. shear strength was dominated by the ice behavior and the leaner. iiitren; brittleies 311in $10 ‘rzzen sci 'Bih caus :Eik Sires ”9T tens} M"liar - I;- I 1.3) Drczl material was more brittle. When the silt concentration was higher than 50%, the stress-strain curve showed a strain hardening character. Temperature Effect: The temperature at which frozen soil is tested has a large influence on frozen soil behavior. A decrease in temperature will generally increase the strength of frozen soil and will also increase its brittleness. The flow rates at lower temperatures are always slower than those at higher temperatures for the same frozen soil type. The embrittlement effect of temperature, ivhich causes a large drop of strength after reaching the [weak stress, will also cause an increase in the compressive ()iver tensile strength ratio according to Sayles and Haines ( 1974), Haynes and Karalius (1977), and Haynes (1978). This irnplies that coldness enhances the ice cement bond between soil particles and it has a greater effect on compressive Strength than tensile strength. The colder the temperature, ‘tt1e more rigid the ice cement bond. Therefore, when the ice Cement bond is broken,soil strength drops rapidly. The relation between strength of frozen soils and tem- Derature is not a simple curve. A plot of temperature versus the unconfined compressive strength is shown in Figure 1.3. In order to approximate the effect of temperature on mechan- ical properties of frozen soils, two methods have been pro- posed for computation. Mitchell, et a1. (1968), and Andersland and AlNouri (1970), proposed a relation based on fi ‘ d 1 .3: $57.35. the theory of rate processes. An empirical power law was first suggested by Vialov, et al. (1962), later used by Sayles (1966), Sayles and Haines (1974), Haynes and Karalius (1977), Haynes (1978), Parameswaran (1980), and Bragg and Andersland (1980). The former appears to apply mainly to clays. For coarse-grained frozen soils such as silts and sands, the power law was found to be more convenient (Ladanyi, 1981). Stress-strain - Time Relationships in the Uniaxial Stress State: Two types of experiments have been used by inesearchers to measure creep behavior of frozen soils. Nith tzemperature held constant, the constant stress test will generally show a strain-time curve as in Figure 1.4(a) and a :S‘train rate-time curve as in Figure 1.4(b). These curves can be divided into three stages: primary, secondary, and tertiary creep. During primary creep the strain rate drops rapidly until the minimum rate is reached. During secondary <:r~eep, the strain rate remains constant until tertiary creep Starts. Nhen tertiary creep starts, the strain rate ilwcreases rapidly, and this is generally considered as failure. In a constant strain rate test, the strain rate and temperature are held constant, and a stress-strain curve is Plotted. Typical stress-strain curves for unconfined com- Dression tests are shown in Figure 1.5. These curves usually show two peaks. The first peak occurs at about 1% . .1 EEK 11131.1 I l" articles. ‘nttion be given Sl’al litltum gtr V699 test he ratio ( Clears to {0'91 one 1; Site MI '1‘ tlcbtaln t all 15 to Silain r515 axial strain, and the second peak will deveIOp at a larger strain, the value dependent on the strain rate. The first peak involves rupture of the ice-cement bond between soil particles, and the second peak represents the ultimate friction between soil particles. A correlation between the peak stress observed at a given strain rate in a constant strain rate test and the minimum strain rate at a given stress in a constant stress creep test was observed by Mellor (1979) and Ladanyi (1981). The ratio (d/é)max under two types of test conditions appears to be the same according to Mellor (1979). There- 'fore, one can put the results from two types of tests on the ssame plot without having large errors. The most popular way t;o obtain the stress-strain rate relationship for frozen SiOiI is to plot test results on a log-log scale. The stress- srtrain rate relationship was expected to be linear on the lcog-log plot. However, Baker (1979), Parameswaran (1980), ialwd Bragg and Andersland (1980) found a bi-linear relation- !Slwip based on their compressive test results. Bragg and Andersland (1980) reported a break in slope of the strain T‘ate at about 1 x 10'5 sec'1. Parameswaran (1980), stated ‘that the break was not observed by early researchers because most early tests were limited to higher strain rates. Parameswaran's (1980) uniaxial unconfined compressive tests included a larger range of strain rates, from 10'7 sec'1 to 10'2 sec-1. More recently, Eckardt (1981), published test resu1ts or unconfined compre551ve tests in the strain rate "r.i . 1 >"" ... a 3. v A r .3 8.: 2. . 3 u: r. i o I. r f. . l . s o o n . . C. .3 ~ v Ir: a n 3.8. a) O t O b I. .x- a x a .Io A.» v o o .0 -h\ § - I‘d range of 10’7 sec'1 to 10'10 sec". His data has been con- verted and compared with other published data in Figure 1.6. From Figure 1.6, one can clearly see that there is another break in the slope at about 1 x 10'7 sec’I. The curves in Figure 1.6 are similar to the stress-strain rate curves of ice summarized by Nixon and McRoberts (1976) except that the stresses are higher and the strain rates are lower in frozen soils. The tensile strength of frozen soils is much weaker than their compressive strength. Experimental problems with tensile tests have limited the available published data. It is generally understood that the largest contributor to the 1:ensile strength is the ice-cement bond. Interparticle f’riction which contributes to the compressive strength of f’rozen soils has a very small effect on the tensile srtrength. The frozen soil in tension, therefore, is more t>t~ittle than in compression. An ideal elastic-brittle material, failing only by t3r~ittle crack propagation, will have a tensile strength Geight times less than its compressive strength (Griffith, 15374). In frozen soil, this ratio varies from 1 at low Stzrain rates up to 5 at high strain rates (Vialov, et al., 1962; Kaplar, 1971; Lade, et al., 1980; Ladanyi, 1981). In the brittle failure range, where strain rate is high, the tensile strength is much less sensitive to strain rate than the compressive strength (Perkins and Ruedrich, 1973; Bragg and Andersland, 1980). At low strain rates, data from Eckardt (1981) showed that the failure deformation in ten- sion was much smaller than that in compression, but the deformation rate in tension is much higher than that in com- pression. Eckardt's (1981) data show that the initial deformation in tension is less than that in compression, which means that the initial tangent modulus in tension is larger when compared to that in compression. Effect of Hydrostatis Pressure and Dilatancy: In engineering practice, loads on frozen soils often include multiple direction of stresses. Andersland and AlNouri (1970) studied the effect of confining pressure in triaxial (:ompressive tests and found that the creep rate of frozen .s<3il decreased exponentially with increase in hydrostatic srtress. Similar effects were observed by AlKire and l\r1dersland (1973), Chamberlain, et al. (1972), and Sayles ( 1973) at low confining pressure. Ladanyi (1981) has presented a Mohr plot, which sche- rnatically represents the entire failure envelope of frozen ()tztawa sand, as shown in Figure 1.7. Note that the failure Quivelope includes three lines. Line I is the failure enve- ltnpe for ice-cement bonds. It covers the region between the tensile strength of ice and the pressure melting point. Line 11 represents the drained failure envelope for the sand skeleton and ice. Line III is the undrained strength of unfrozen sand. 01-“ :lvb -!I1t‘\" “. a ...,rni ...-i.- 0" Au» .hd «F. i 6:. we I . we. 5- «416 c L. "I has. p .1 . 1 .4.- q. Ah\ .I I :5 l .t .J a v n ... ..I’ in .: 0 N in \r ..v d n it u I 11 5|... 10 The effect of confining pressures on the shear strength of frozen soils is shown in Figure 1.7. The transition con- fining pressure between regions A and B is about 3 to 4 MPa according to Sayles (1973). The transition confining pres- sure between regions 8 and C is about 55 MPa according to Chamberlain, et a1. (1972). The majority of permafrost engineering problems usually includes regions A and B. For other types of frozen soils, similar results have been shown by Conlon (1966), and Loiselle, et al. (1972) for some cemented clays, and Clough, et al. (1979) for silicate grouted sands. The dilatancy effect, observed in regions B and C in FVigure 1.7, has also been noted by many researchers. When a “Ficozen soil is subjected simultaneously to hydrostatic pres- Eitlre and shear stress, two kinds of situations will develop. I'f’ the confining pressure is low, the volume of the frozen 53:)il expands and dilatancy releases some of the stress in tirie ice matrix and soil skeleton during shear. If the con- f’i ning pressure is so high that dilatancy is suppressed, the Silear strength of the frozen soil will be increased. These DWIenomena are similar to the "dilatancy hardening" and "dilatancy softening“ in rock and unfrozen soils according to Ladanyi (1981). The former was observed and discussed by Goughnour and Andersland (1968), Andersland and Douglas (1970), AKili (1971), AlKire and Andersland (1973), Perkins and Ruedrich (1973), Ruedrich and Perkins (1974), Sayles and Carbee (1980), Bragg and Andersland (1980), and Lade, et al. 31 time t 1 a ' U- ‘ G: d? = L :19 .. n ‘qutl 11 (1980). The latter was also found by Chamberlain, et al. (1972) at high confining pressures. 1.3.2. An Engineering Theory for Frozen Soils Constitutive Equation in the Uniaxial Stress State: The creep curve shown in Figure 1.4(a), obtained by constant loading tests under a uniaxial stress condition, is common to a large number of materials including frozen soils, polymers, and many metals at high temperature. The strain at time t is a function of loading and temperature and can be written as €=€.+€ (1'1) I f’we write the instantaneous strain,ei , as 61 = F(O,T) (1'2) c and the creep strain rate, Edit” as EC 35’ C3. = G(O,T) (1.3) 0.1 the constitutive equation of these types of materials can be Written as (Hult, 1966) 12 e s F(o,T) + Iguana (1.4) The instantaneous strain, e1, can be further divided to an elastic portion, ale, and a plastic portion, elp as follows 9 J = eie + JP (1.5) , - (1.6) ivhere E(T) is the initial tangent modulus. For the plastic t)<3rtion, Hult (1966) and Ladanyi (1972) have written alpas a Dower expression ip O k(T) (1.7) ‘ = ark—W) vvriereok is a temperature-dependent proof stress,e,k is an iir‘bitrary small strain introduced for convenience in calcu- léition, and k is a constant which usually is little affected IDy the temperature. Substitute Equations (1.6) and (1.7) into (1.5) and the total instantaneous strain can be written as: 5i = O + 6 (L) (108) E(T) k “U” cw ‘II :ff'l" towiil 3:: Ere: a": isle .‘9: IE A: PI.‘ 13 For small strains, the second term in Equation (1.8) is small and the instantaneous strain can be approximated as i . ‘57 (1.9) where E' is the pseudo-instantaneous elastic modulus. If the strain is large, the first term is small and the instantaneous strain is: k sink-9') (1.10) °k according to Hult (1966). Creep Laws in the Prefailure State: As mentioned ea rlier, the creep curve in Figure 1.4 includes three parts: F>r‘imary, secondary and tertiary creep. Since tertiary Cirreep is considered as post-failure behavior, only F>r‘imary and secondary creep are of interest in engineering F>rfiactice. The equation for computing creep strain is called the creep law. The most popular creep law in frozen soils, <>r‘iginally proposed by Vialov, et al. (1962), can be written iis follows (Landanyi 1972; Ladanyi and Johnston 1974; AI‘Hzlersland, et al. 1978; Eckardt 1981): a: = We)" 1" where azis an arbitrary strain rate selected for convenience , nfifnl-‘ ‘1 " lirU :ar 1'”? , cw qr -\.P u, . .C An: r: use: II-‘n 03r- 4‘ Jar: :I.‘ . ,o 1 r \- B. (..1 14 in computation, 0C is a temperature-dependent proof stress for normalization of dimensions, n is the exponential parameter for stress, and b is the exponential parameter for time. When b <1, equation (1.11) is for primary creep and when b = 1, equation (1.11) represents secondary creep. Another creep law, originally proposed by Nadai (1938) and used for frozen soils by Andersland and Akili (1967), Mitchell and Campanella (1968), Goughnour and Andersland (1968) and Andersland and AlNouri (1970), has the form c _ - . o e -usmh(§)t (1AM The parameters u and % in Equation (1.12) are similar to éc and ac in Equation (1.11). Nevertheless, if one plots Equation (1.11) and Equation (1.12) on a log-log scale, Equation (1.11) will be a straight line while Equation (1.12) is nonlinear. Another creep equation, which is bi-linear in the log- lil interface occurred. Similar studies on bond behavior at the pile/frozen 1972; Nixon SC>i 1 interface are many (Johnston and Ladanyi, arici McRoberts, 1976; Nixon, 1978; Parameswaren, 1978, 1979; Weaver and Morgenstern, 1981; Morgenstern, et al., 1980). The current understanding of bond behavior is discussed in the following sections. Load-displacement Curves from Pull-out Tests: Alwahhab (13383) conducted a series of constant displacement rate Pull-out tests on an embedded smooth steel bar in frozen (I‘D ’F 01a ~ DC 35nd 19 soils. The test results showed two types of load-displace- ment curves (Figures 1.10). For temperatures lower than -1ooc,a slip combined with a rapid drop of loading occurs after the peak strength is reached. For temperatures higher than -6°C, the load gradually decreased after the peak strength was reached. Figures 1.11 and 1.12 show test results at -6°C and -2°C with different loading rates. Note that slip occurred at higher loading rates at -6°C but not at lower loading rates or at -2°C. Similar effects were observed by Paramewswaran (1978) in model pile load tests at -6°C. His data showed that the pile load reached a peak and then dropped quickly, indicat- ing that the bond between pile and frozen sand had broken. TWie shape of the post-peak curve depends on the pile mate- rfiial and rate of loading. In a few cases with painted steel pipes, when the rate of loading was greater than 0.02 that/min, the load dropped abruptly after the peak, indicating a (:lean shear at the pile/frozen soil interface. For engineering practice, only the strength and behav- iC>r~ in the prefailure state are of interest. Nevertheless, t'Tee post-failure modes are also important. A rapid drop of b0nd strengthwill cause a sudden collapse of structures with no time for escape of occupants or repair. _ ,l . lClI) m (TESS-E u.\1 20 Ultimate Bond Strength at the Pile/Frozen Soil Inter- face: Two different methods are available for describing the ultimate bond strength at the pile/frozen soil inter- face. One employs the Mohr-Coulomb relation for frozen -soil; the other uses a power law relationship. When expressed by the Mohr-Coulomb relation, the bond behavior has the form Tit = C21: + otan 1b“; (1.19) where o is the normal stress on the shear plane, tit is the long-term shear strength, ¢£t is the long-term friction angle, and C21: is the long-term cohesion. Weaver and Morgenstern (1981) argued that the normal Stncess on the adfreeze shear plane is small; thus, the fric- ti<>nal component will be generally insignificant and may be neglected. Weaver and Morgenstern (1981) related the acif’reeze strength ea to the long term cohesion as ta .3 Inc“; (1.20) Where in is a constant depending on the roughness of the pile Sllr‘face. A summary of long-term cohesion values for frozen 8Oils and the m coefficient are listed in Tables 1.1 and 1 .2. The value of m varies from 0.6 to 1.0 in Weaver and MC3"‘genstern's study. 7 Since the strength of frozen soil is dependent on the aDDlied strain rate, it was reasonable to assume that the ultimate bond strength has a similar relationship. an; nyfl 5 n. a. 0.4 u b It: L'I‘ e 1 .- i TIO‘.‘ )1 files tiles. '. 1. OF: .1: . b 191’ 1|): Eu s in! 21 Parameswaran (1978) plotted the peak adfreeze strength versus the applied displacement rate on a log-log scale. The results showed that the power law could also be applied to the bond behavior on the pile/frozen soil interface. The relation can be expressed as 5=Atn (1.21) where d is the displacement rate, r is the shear stress on the pile/frozen soil interface, and A is a constant. The creep parameters, n, given by Parameswaran's study are listed in Table 2.3. Based on a comparison of different pile surface types, Parameswaran (1978) concluded that the maximum adfreeze bond strength developed with uncoated wood Di les. Concrete piles developed adfreeze bond strengths lower than wood but higher than steel. A surface coating greatly reduced the adfreeze bond strengths. Parameswaran (1979, 1981) has also compared the ulti- mate bond strength on the pile/frozen soil interface with that on a pile/ice interface. The results showed that the ultimate bond strength on the pile/frozen soil interface was "HJ<:h higher than that on the pile/ice interface. It tNisures the contribution of surface friction on the ultimate bCJnci strength. Similar results were also shown by Alwahhab (1983L Andersland and Alwahhab (1982) studied the temperature effects and loading rate effects on the ultimate bond Strength. The results showed that the ultimate bond r1.” .0 ‘ \ r9 ~4 of _,l s 1.97. S P'fid 1' ;“ r2" 9. al‘ .c'l' .ECCWC ..4 11) Li a \ F O I r 1 '0 the 1'” iii Dé‘r 22 strength increased linearly with decrease of temperature at slower loading rates. At higher loading rates the ultimate bond strength increased more rapidly with a decrease in tem- perature than at lower loading rates. At higher loading rates, the ultimate bond strength increased more rapidly with decrease in temperature from 0°C to -5°C. At temperatures colder than -5°C, the increase in bond strength was similar to that of slower laoding rates. Creep at the Pile/Frozen Soil Interface: Johnston and Ladanyi (1972) found that pile displacements under a sus- tained pull-out loading showed three creep states (primary, secondary, tertiary). the same as in the uniaxial compression teast for frozen soils. In a simplified analysis, Johnston arid Ladanyi proposed a creep law which related pile displace- ment rate with bond stress on the pile surface. This creep liaw was later modified by Nixon and McRoberts (1976), Nixon (‘l978), Morgenstern, et al. (1980), and Weaver and Morgenstern (1981). In general the creep law for bond behavior has the form (1.22) where the pile displacement rate 5 is divided by the Di 1e radious r for normalization and is related to the aDplied shear stress I by the creep parameter n. The 11 :terlals ‘h: ="1Ed .-. ’fiural 3. >r‘Ladany field test Ifi about The dial str erg, 311 $011 23 The value of n varied for different pile surface materials and different temperatures. Parameswaran (1979) showed that n was 6.7 for a painted steel pile, 8.1 for natural B. C. fir and 9.1 for concrete at -6°C. Johnston and Ladanyi showed that n was between 7.5 to 8.5 in their field tests. Alwahhab (1983) showed for steel bars that n was about 2.75 at -10°C and 4.75 at -15°C. 1.3.4. Flexural Creep Behavior of Structures in the Uniaxial Stress State The creep behavior of structural elements in the uni- axial stress state, such as those in beam, column, and truss members, can be analyzed by one-dimensional theory. Closed fornl solutions of some one-dimensional creep problems have been found. The discussions herein are limited to only the creep of beams. The flexural creep behavior of lead beams was studied as early as 1933 by MacCullough. He discussed 60d verified Bernoulli's assumption that a plane section remains plane throughout the deformations. Topshell and JOhnson (1935) discussed the stress distribution over the ‘3POss section in the steady-state creep condition. Popov (1949) solved numerically the bending of beams for both '3rimary and secondary creep. The creep deflection of a beam isubjected to both axial and lateral loads (beam-column) was also analyzed by Hu and Triner (1956) and Lin (1958). A method of analysis which used an analogy between the elastic strain and the creep strain rate was proposed by 24 Hoff (1954). With the method of elastic analogy some structural creep problems can be solved analytically. A beam with a bisymmetrical cross-section subjected to pure bending in steady state creep was considered by Hult (1966). Assuming that the bending moment acted in a plane of sym- metry and the center line of the beam deformed into a plane circular arc (Figure 1.13[a]), Hult derived the equation for o the stress distribution along the cross section x, 1 °x‘7M"Z' (“son 2 (1.23) n in which 1+1/ ln = IAIZI " dA and M is the applied moment, Z is the distance from the neu- tral axis, n is the creep parameter for the power creep law. When n equals one, I1 is the area moment of inertia of the cboss section. For a rectangular cross section 1/ 2nB ( )2+ n _ = H 1“ 1+2n (1.24) N|—- The shape of stress distribution over the cross section is ‘dependent on the creep parameter n. When n equals one, the Stress distribution is triangular. When n equals infinity, the stress distribution is rectangular (Figure 1.13[b]). PI-A 1 1’35 ””w C ”'7'DC bit-",1 1. J (I) 25 Bragg (1980) has attempted to analyze a frozen soil beam with different creep rates in tension and compression. In the analysis, he used different values of the proof stress but the same value of the power parameter n for ten- sion and compression. This led to the result that the neu- tral axis of the beam in his analysis was closer to the compression side and the tension strain was larger than the conuaressive strain. The Opposite would result if the cor- rect: power parameters were used. 1w3.5. Analysis of Structural Creep Behavior in a Multi-axial Stress State For structures subjected to creep in a multiaxial stress state“ analytical solutions are very difficult to find. Without the use of digital computers, only a few cases of structures in multiaxial creep have been considered. Popov (1947) has analyzed the creep behavior of turbine disks at high temperatures. Some other examples such as rotational disks, circular shafts, and hollow cylinders were reported by Mendelson (1959) and Hult (1966). Since high speed com- DUters became available, a large amount of multiaxial structural creep problems have been solved by finite element methods. Because of the scarcity of experimental data on frozen sOil properties under multiaxial stress states, to solve mUltiaxial structural creep problems by finite element method, it is necessary to generate multiaxial creep l M 1112-11336 1.1. E «l c , . tat-101 RN\ nah 26 relationship from uniaxial creep data. Homogeneity and iso- tropy were usually assumed in the analysis (Tompson and Sayles, 1979; Klein and Jessberger, 1979; Klein, 1982). For a homogeneous isotropic material, the stress-creep strain rate relationship is °c 3 g: wherwe égj is the creep strain rate tensor; Sij is the stress deviatoric tensor; and 0e and E: are the "effective stress" and ‘the "effective creep strain rate", respectively, and they can be expressed as _ 3 % ° _ 2 0C cc V2 In the uniaxial stress state, Equations (1.26) and (1.27) are reduced to T“Us, uniaxial experimental data can be used to predict ‘zreep strains in the multiaxial stress state by using the "effective stress" and the "effective creep strain rate." Rh nt- -\4 14s A H b 'Eiiew 27 Analysis of structural creep problems using more complicated yield criteria such as Drucker-Prager, Mohr- cOulomb and Tresca criteria were reported by Zienkiewicz and Cormeau (1974), Zienkiewicz (1977), Zienkiewicz and Humpheson (1977), and Mron and Sharma (1980), etc. To define these yield criteria more experimental data are needed. For analysis of frozen soil structures, they do not seenl suitable to use due to a lack of information on the material parameters involved. Therefore, they will not be reviewed here . Besides the stress-creep strain rate relationship in multiaxial stress state, the solution technique used to solve the structural creep problems are also important. Most researchers used incremental approaches in the time domain (e.g., Greenbaum and Rubiustein, 1968; Zienkiewicz and Cormeau, 1974; Argyris, et al. 1978; Klein and Jessberger, 1979; Klein, 1982; etc.). The solution tech- nique usually started with the assumption that the total Change in strain during a time interval was the sum of the changes in elastic and creep strains (Greenbaum and RUbiustein, 1968), i.e.. {A8} = {Ase} + {use} (1.28) ‘VDEre the superscripts e and c denote elastic and creep respectively. The change in the elastic strains is thus: (Ase) = (As) - {Aecl (1.291 28 The incremental stress is then related to the elastic strain by the generalized Hooke'. law {A0} = [E]{Aee} (1.30) Substitution of Equation (1.29) into (1.30) yields the rela- tion between the incremental stress, total strain, and creep strain: {A0} = [mud-{115)} (1.31) Using Equations (1.25) and (1.31) with the relationship between strain and deformation, a system of simultaneous equations can be formed by applying the theorem of Stationary Potential Energy. With step-by-step calculations, the stresses, strains and displacements in each time step can be solved. The solution technique is as follows: (1) At time t = 0, the elastic stresses are determined. These stresses are assumed to remain constant during a small time increment, At, and the incremental creep strains are calculated from Equation (1.25). (2) The creep strains are converted to “pseudo-force" (Zienkiewicz, 1977) and substituted into a structural stiffness matrix to find the total change of the nodal point displacements. Then, the total changes in strains are calculated from the displacement-strain relationship. 29 (3) Finally, the incremental stresses are obtained by (1.31) and are added to the previous stresses to yield the new stress distribution. These new stresses, then, are used to calculate the incremental creep strains in the next time step, and the procedures in (2) and (3) are repeated. The preceding approach may be called the straight Euler metfliod. No iteration is used in solving the system of equa- ticnis in each time step. It has the disadvantage that munerdcal instability would occur when the time step is too large. The condition of numerical stability has been discnissed by many authors (Zienkiewicz and Cormeau, 1974; Cornueau, 1975; Argyris, et al., 1978; etc.). A rule of thumb is that the incremental creep strain should not exceed one- half of the total elastic strain (Zienkiewicz, 1977). Another incremental approach, the tangent method, was introduced by Cry and Teter (1973) and Zienkiewicz (1977). The method computes the incremental creep strain at the middle of each time step. It has the advantage that it is u"conditionally stable. Nevertheless, a part of the pseudo- f0rce matrix which was generated from incremental creep Strain in each time step has to be added to the structural Stiffness matrix. A penalty has to be paid in that the structural stiffness has to be reformed in each time step. OCcasionally, the structural stiffness matrix will become nOn-symmetric. This makes the analysis very complicated and difficult to apply. .41 I . 101M114 c. D.» IN.» LP . filth 1' 30 1.3.6. Interface Models for Finite Element Analysis The bond behavior between two materials, such as the pile/frozen soil interface discussed in Section 1.3.3, is very complicated and special finite elements are needed for modelling. A bond-link element was used to model the bond behavior in reinforced concrete by N90 and Scordelis (1967) and Nilson (1968). The bond-link element contains two linear springs in two perpendicular directions. 'Hue element has zero (initial) length and zero (initial) widtfli as shown in Figure 1.14. The two linear springs are aSSLuned to be independent of each other. The relative dis- placements between the nodes, which are attached to the interfacing materials, are transformed to the nodal displacements by a transformation matrix. Another type of interface element, introduced by Goodman, et al. (1968), was used to model joints in a rock mass. This joint element was subsequently modified by Ghaboussi, et al. (1973) with isoparametric formulation. As Shown in Figure 1.15(b), the joint element has a thickness t and uses relative displacements as independent degrees of freedom. The nodal displacements are transformed to the relative displacements by a transformation matrix. The strains are assumed to vary linearly along the length of the element and remain constant along the thickness. The strain- diSplacement relationship in the element coordinates is as fOllows (Figure 1.15[a] and [b]) Here 0 31 6hi , (14:) 0 (1+5) 0 6 {5n} = — si (1.32) 6S. 2t 0 (l-n) O (1+n) on. J ésj where an and as are strains; éni’ ési’ onj,and bsj are rela- tive displacements. The subscripts n and 5 denote directions perpendicular and parallel to the element. The stress-strain relationship is assumed as on = Cn 0 an (1.33) as 0 CS ES wherwe o and a are stresses, and Cn and CS are stiffness n s constants perpendicular and parallel to the element. Using the theorem of Stationary Potential Energy, the stiffness matrix for the element can be derived. There are many applications and modifications of joint Elements to solve interface problems (Zienkiewicz, 1973, 1977; Pande and Sharma, 1979; Heuze and Barbous, 1982; Desai, 1983; etc.). They do not include creep response on the bond interface. There also are other types of interface models in finite element method. Examples which model the interface behavior by use of a Langrange multiplier have been reported by Herwmann (1978), Haber and Abel (1982, 1983). However, among all the interface models reviewed, the joint element seems m0St suitable for inclusion in a straight forward stiffness ArnA .- Fa, ;n\ hard-.1 32 formulation of the structural problem. It will be used herein with appropriate modifications to include the creep response described in Chapter IV. 33 Table 1.1. Summary of Long-Term Cohesion for Frozen Soils (kpa) (After Weaver and Morgenstern, 1981). Temperature Ice1 Ice-Rich Varved Ice-Poor (°C) Silt2 Clay2 Fine Sand2 —0.35 55 180 230 -1.15 100 260 270 -1.2 160 -1.8 200 -4.0 300 -4,2 300 470 450 1Voitkovskii (1960). 2Vialov (1959). 34 Table 1.2. Summary of m Coefficient (After Weaver and Morgenstern, 1981). Pile Type m Steel 0.6 Concrete 0.6 Timber (uncreosoted) 0.7 Corrugated Steel Pipe 1.0 Table 1.3. n Value for Different Types of Piles at -6°C (After Parameswaran, 1978). Material n B. C. Fir 4.48 Concrete 4.63 Painted Steel 6.02 Creosoted B. C. Fir 5.37 Unpainted Steel 9.66 Natural Spruce 10.23 H-Section 10.08 35 pp~v “T’” 46m (a) ,9 Soft Soil :5 Opdn Excavation Frozen Soil \‘ i Opqn Excavation (b) Frozen Wall Freeze Pipes (0) Figure 1.1 (a) Open excavation supported by straight frozen walls. (b) Open excavation supported by a curved frozen wall. (c) Tunnels supported by frozen walls. 6 ANExxzs: S unuuucL 19”," 4 3 H a» . X< x.....,w.,~ «(IV 36 IO— 9 >- 8 >- £1 = 2.66xlO-LIrnin“l T = —12.0 0 7 '- 3 C \ of‘ _ _ 2: £1 = 1.33x10 4min 1 6 E -12.o3°c (0 U) CD 34 5 +3 (0 H m :2 4 q: x m CD 91 3 2 \\\\\ - - El = 2.66xlO “min 1 T = -3.85°c 1 1— 0 I I J i l J l o lo 20 3o 40 50 60 70 Percent Sand by Volume (%) Figure 1.2 Volume concentration of Ottawa sand peak strength (after Goughnour and Andersland, 1968), D\a.if| w ‘ Eli's-I.- c: w "b” alto-E(r\ . 2 «5:11» &wu NE\22 37 C .9 &> N 60 r 50_ ‘0 O ~S 4o - Ottowa sand Unconfined compressive strength, MN/m2 O . -SO ~100 -150 0 Temperature, C Figure 1.3 Temperature dependence of unconfined compressive strengths for several frozen soils and ice (after Sayles, 1966). b :« EL on». 1111.11 I I. IIIII|.I.I II I . we obs-\L CursLunw .IIIIIIIK Ar \ oerc ‘« 38 o = constant A III Tertiary creep U c -H 3 1 I U m | I I I II Primary I Secondary creep I C reep I I 4;; (a) Time t A I I 1 I «a l I o P | 3 .5 I l S u l I U) I 1 I III I II I l \ (b) Time t Figure 1.4 Constant-stress creep test (a) basic creep curve (b) true strain rate vs. time. 5: v ... ..i 39 .Aowma .vcwamhouc< was wwwnm nmpmmv mvmmp CofimmwumEoo vmcflHCoocs pom mm>u50 cfimnvmummmupm HwOHahe m.H mpzwflm :3 u .523 RUE m n o m v m p p p p p _ com 0 oooH m. I Cu 0 1. J a m: 8 00mm 3 9 d S Anumm nuoa x mm.v u m>m w COON HH H..owm muoH x mo.m u o>m w Hnowm :uoH x mm.m u m>m u U 00H: n musumummeme r oomm D Ixhbb,h.v . hi hi. *..hi h ~D...’ b ‘ ~:¥1|l|h~ubuh P O cry-kc I \ Rogue H Ir.l .lrlll I. vIS'IIIII III I? [I «:29 N » :rwh REJECECL .....n II \IIIII ll 'l'I-Irlll » Cwil.Cp~.mn.~ nuns-x Int~\n-\U. . . Nbbb‘b'tl 5 IL K hub h IPIrL [bill‘Ku 515 E1) H. sv~ 4O .mnospzm 92mpmmmfln an cmpuoamg mmpmu cflmnpm Cowmmwpasoo ho ComfithEoo w.a mmswwm Afilwav U wUQH CMGHUM « mlawqa:. « a A q :lO—M: . u. _ . mlO—..n__q u . A _ mlquu-d - . q . thwn—fiqd. . u . .mlOH_:_ _ a. . mlOH 1.1. . _ TI- 1 1 a n com- A emfin 1 w m oomnm H. Oomfl 1 Om w|||o l h u f Admch uommxom 1 I Aommav vcmHmnmoca can ounum .n uuuuuuuuu I I ' AommH. :mumzmmawuum I n “oomflc cfinucmmm can mmaxmm . u WL rtbrp _ p _._-.- p - rpthP r 7:..- . . —:b-P. - p —-:.- p . —.-b_- u OH NoH moH JOH mod (13d) 0 588135 41 Amuma .mmahmm ccm .mmma ..Hw pm :fimempemno umpmwv gnaw mzmppo Cmuopm pom macaw>Cm mpsfifimm macs; may Ho coflPMpCmmthmh Ofipmswnom u.H opsmfim wusmmmum oflumum0u©>x umcc: wcwuamz musmmwum _ All| a VA 0 LA m vx th > tf3 6 = constant J ° I t all c I? of [I of I] Figure 1.8 Time dependent failure envelopes (after Ladanyi, 1972) 43 constant V U le——-H >I.< a $4 f Figure 1.9 Straight-line approximation of time dependent failure envelopes. a ‘ . “cuuay.~ ct F~a 44 ” T — -2:c, s-27, b = 26.3 lbs/min T = —6 g, S-3, p = 29.7 lbs/min T = -lOOC, S-2, p = 32.5 lbs/min T = -lSOC, S—4, p = 30.7 lbs/min 3-‘ T = ~2ooc, s-s, p = 32.7 lbs/min T = -26OC T = -26 C, 5‘7, b = 39.7 lbs/min ‘l,,,.—T = -2o°c I A ll 3‘” .. "-1 x s: we CL _ 0 Smooth '3 steel bar 0 t: Frozen " sand 0.62 \ 1r 5 . ' u _ - 6.. l J. I 20 Displacement (lo-Bin.) Figure 1.10 Load-displacement curves for "pull out" tests at different temperatures (data from Alwahhab, 1983). 45 3 s-3 b = 29.7 lbs/min, 3n = 6.967 x 10‘“ in/min 5—10 6 = 186.7 lb/min, 6n = 1.12 x 10’3 in/min s—13 p = 2133.3 lb/min, 6n = 3.0 x 10‘2 in/min s-2o p = 4.4 lb/min, 6n = 1.86 x 10*“ in/min U) CL 0H .x V '0 a. I‘ €3"( -2 + 'U m 0 ,q 625 sand ‘fl’o. ...—8"13 Smooth steel bar Frozen //// 'U ‘—-(: Displacement (lo-Bin.) Figure 1.11 Load-displacement curves for "pull out" tests at different displacement rates at -6 C (data from Alwahhab, 1983). 46 0.7‘ \\\\\\\\T3 s-24 6 = 2.46 x 10'“ in/min S-2S o = 2 x 10‘3 in/min 0.6“ 8-26 § = 5.7 x 10"6 in/min S-27 5 = 3.7 x lO'Sin/min ‘,s-25 k-6—4 [J _ Smooth 0-5 steel Frozen bar \ I Y 1 T I O 5 10 15 20 25 Displacement (lo-Bin.) Figure 1.12 Load-displacement curves for "pull on " tests at different displacement rates at -2 C (data from Alwahhab. 1983). 47 (a) 22 2 6 M/BH2 4.67M/BH2 14.20M/BH2 u M/BH // // (J Figure 1.13 (a) Pure bending of a bisymmetric beam. (b) Stress distribution in bending of a rectangular beam for various values of the power parameter n. 48 d Scordelis 4 B d lind element developed by Ngo an ‘ .l on - Figure 1 (1967). ’ll 49 1; Upper Continuum Element Joint Element Lower | l Continuum Element X (Global Coordinate) (a) rln its 9 (Local Coordinate) m N er two (.1 X n (Global Coordinate) (b) Figure 1.15 (a) The joint developed by Ghaboussi et al. (1973). (b) The joint element in local coordinates. CHAPTER II MODEL BEAM TESTS ON FROZEN SOIL 2.1. General Experimental results for frozen sand-ice beams are presented in this chapter. Materials used, sample prepara- tion, equipment and test procedures are described. Two beam types, plain and reinforced, were tested at -6°C and -10°C. The beams were loaded in such a way that they were subjected to pure bending in their middle portion. Deflections were measured. Experimental results have been plotted in both normal and log-log scales. A comparison and a discussion of the results are included. 2.2. Materials and Sample Preparation For convenience, commercially available silica sand produced by the Hedron Division of the Pebble Beach Corporation of Nedron, Illinois, was selected for these tests. The sand gradation was uniform with all material passing the number 30 U.S. standard sieve and retained on the number 140 sieve. The specific gravity was 2.65 and the coefficient of uniformity was approximately 1.5. A sand volume concentration of 64 percent was selected so as to relate this work with published tensile and compressive test data. This sand volume fraction is well 50 A t nh'ill ‘ .‘l 5.4.4 3» AJ AME Sunni later 7‘; . VCUPQ This 51 above the critical volume concentration of 42 percent deter- mined by Goughnour (1968). The sand volume fraction was determined by preweighing the correct amount of sand and water. To avoid possible effects of any solutes, distilled water was used in preparation of the frozen beam. A beam size of 2.75 inches wide, 3.25 inches deep, and 40 inches long, satisfied the considerations described by Leonards and Narain (1963)*. All samples were prepared in a wooden mold with thicknesses of one-half inch in most of the mold parts. The mold was cleaned before each sample prep- aration and a sheet of Saran Wrap was applied to the mold surface to aid in sample removal after freezing. Two 2.75"x 1" x 0.5" steel blocks with three lines of %-inch nails were placed into slots at the bottom of the mold for beam supports. To prepare the sample, a small amount of distilled water was poured into the mold, then, the sand was slowly poured into the water to make a one-half inch thick layer. This allowed trapped air bubbles to escape from the sand. Sample compaction was achieved by tapping the surface of the sand. This procedure was repeated until the mold was filled or the preweighed sand and water were totally used. Two *The considerations were: (1) span to depth ratio greater than six; (2) easy manipulation by one man without excessive handling stresses; and (3) deflections large enough to be measured precisely with transducers. KS. k‘q“ L'iV" ale we 2.3. 2.3.1 52 blocks, similar to those for beam supports, were positioned two inches from each end of the mold for load supports. Thermistors were placed near each end of the mold within the sample to monitor temperatures. The mold and sample were then frozen and maintained at the test temperature for more than twelve hours. When the thermistors showed that temperatures at each end of the sam- ple were the same, it was assumed that the temperature was uniformly distributed along the sample. The sample was then removed from the mold, wrapped with protective membranes, and placed in the cold bath of the test unit. For reinforced beam samples, a 3/16-inch cold rolled steel bar with smooth surface or a 3/16-inch steel bar with a 1/16-inch lug at 3/4 inch from each end was placed one-half inch from the top surface of the mold during sample prepara- tion. The surface of the cold rolled steel had a roughness factor of 625% according to Alwahhab (1983). The lugs had a 90-degree angle from the surface of the steel bar. 2.3. Equipment and Test Procedures 2.3.1. Equipment A test frame with a coolant tank 50 inches long, 16 inches wide, and 16 inches deep was used to conduct the experiments as shown in Figures 2.1(a) and (b). The coolant tank. insulated to reduce heat intake, had two holes on each end for coolant inflow and one long slot at the middle of the side for coolant out flow. Coolant was supplied by a 53 refrigeration unit and pump for circulation. The coolant mixture was prepared by mixing water and ethylene glycol (anti-freeze) on a 1 to 1 basis. The beam sample was supported by two hinge type beam supports as shown in Figure 2.2. Beneath these supports were two 20-inch long, 2-inch wide, and 2-inch deep steel bars for load transfer to the larger test frame. These two steel bars were connected by two 3/8-inch rods to avoid pos- sible sliding. The radius of the slots was made larger than the radius of the rods to allow for a small longitudinal beam movement. Loading was applied at both ends of the beam by a dead load and hanger system as shown in Figure 2.1(a). A hinge system was also used to apply the load to the beam. The frozen sand beam was protected from contact with coolant by membranes. The membranes were wrapped over and pinned on top of Styrofoam insulation placed above the sur- face of the coolant. The piece of Styrofoam had several openings to allow for hangers, transducer cores. and ther- mistor wires. The beam temperature was measured by thermistors at each end and monitored by a Hewlett-Packard digital logging multimeter (Model 3467A). The transducer cores were mounted 2.5 inches above the top of the beam. Beam displacement was measured at the middle and one-quarter points along the beam length, on both sides, by six linear variable displacement transducers and recorded by a Sanborn two channel recorder (Model 77023) with preamplifier (Model 8805A) and a Sanborn 150 four channel recorder. 54 2.3.2. Test Procedures One hour before the test started. the temperature of the coolant was set at 5 degrees Celsius below the test tem- perature to allow for heat loss from the test frame and tank during placement of the beam in the test system (Figure 2.1[a]). The beam support and the hangers were adjusted to the proper position. Membranes were put into the empty tank on top of the support system. After the sample was removed from the mold, the following procedure was followed: (1) Place the sample in the freezer without mold and set the temperature of the freezer at five degrees below the test temperature for one hour. At the same time turn on the recorder for warm up. (2) One hour later place the sample into the tank at the correct position for loading and attach each transducer core to the beam. (3) Place the Styrofoam insulation on top of the beam, pin protective membranes to the Styrofoam, and lower both load hangers until they touch the beam. (4) Fill the tank with coolant and circulate until a uni- form temperature is achieved. Set the temperature of the refrigerated coolant to the test temperature. Monitor beam temperatures with the multimeter. (5) Connect the transducer circuits to the recorders and adjust to a zero reading. 55 (6) Load both hangers with the same dead weight of lead bricks and lower both hangers at the same time to apply the first load increment to the beam. (7) Monitor beam behavior as the test progresses with deflection recorded by recorders. (8) Each load increment was continued for at least five hours after steady-state creep was reached. New load increments were applied by placement of more load bricks on the hangers. The test was stopped when the deflection at mid-span exceeded 0.4 inch (the capacity of the test unit was about 0.5 inch) or when rupture was detected. (9) Next the coolant was drained from the tank. Membranes and test apparatus were removed from the sample. The sample was oven dried at 115°C to determine the dry weight of sand. (10) The recorder strips were marked and filed until data could be transcribed to data sheets. A temperature five degrees below the test temperature was used before the test started so as to compensate for a temperature increase during mounting the beam in the test apparatus. This procedure helped bring the beam to a uni- form temperature more quickly. The test temperature was monitored and recorded as the test progressed. When the test temperature rose or dropped by 0.2 degree. the temper- ature of the refrigerant coolant was adjusted to compensate. . . All. A {L 56 2.4. Experimental Results and Discussion 2.4.1. Experimental Results Six frozen sand-ice beams were loaded by a series of step loadings in this test program. The load increment for each step was about twenty-six pounds. Samples 1 and 2 failed because of membrane leakage. Samples 3 and 4 were plain frozen sand-ice beams and were tested at an average temperature of -10.3°C and -6°C, respectively. Sample 5 was reinforced with a 3/16-inch steel rod placed one-half inch below the top of the beam and was tested at an average tem- perature of -10.2°C. Sample 6 was reinforced the same as sample 5 except that the steel rod had lugs 1/16-inch high by 1/2-inch long at 3/4 inch from both ends and was tested at an average temperature of -10.1°C. All samples contained the same sand volume fraction, close to 64%, with beam dimensions of 40 inches long, 3.25 inches deep, and 2.75 inches wide. All beams were supported 10 inches from both ends and loaded 2 inches from both ends, as shown in Figure 2.3(a). Beam deflections were measured at mid-span and five inches from both supports on both sides. The transducer cores were mounted 2.5 inches from the top of the beam, as shown in Figure 2.3(b). Test results for sample 3 at about -10°C are shown in Figure 2.4. Note that most of the deflection curves include a curved portion and a straight-line portion, representing both primary and steady-state creep. The test results for sample 4 and -6°C are shown in 57 Figure 2.5. The deflection curves in Figure 2.5 have the same shape as that of sample 3 but the deflection for each load increment is larger. The computed deflection rate for the straight line portion of samples 3 and 4 are listed in Tables 2.1 and Table 2.2. The deflection rates versus load data from Tables 2.1 and 2.2, plotted on a log-log scale in Figure 2.6, show linear relationships. For the reinforced beam, Figure 2.7 shows deflection versus time curves similar to those of sample 5. The primary and steady-state creep was observed for all the deflection curves. In Figure 2.7, the beam was initially loaded at 96.16 lbs., and rupture occurred as the load was increased to 459 lbs. Figure 2.8 shows the deflection versus time curves for sample 6 which was reinforced with a 3/16-inch steel rod with 1/16-inch lugs at both ends. The initial loading was 148.05 lbs., and the rupture loading was 511.05 lbs. When deflection rates for steady-state creep were plotted versus loads, both samples showed a bi-linear relationship on a log-log scale, as shown in Figure 2.10. The slope at the left portion of the bi-linear curves was larger than that at the right portion of the bi-linear curves, which means that the deflection of the beam increased at a slower rate at smaller loadings, then changed to a faster rate when a crit- ical load was reached. The critical load was about 180 lbs. and 280 lbs. for samples 5 and 6, respectively. Note that the two bi-linear curves in Figure 2.9 were almost parallel 58 with each other in both portions. When the bi-linear curves are compared with the linear curve of sample 3, note that the left hand portion of the bi-linear curve has the same slope as that of the plain beam, shown in Figure 2.10. The test of sample 3 was stopped due to excess deflec- tion at 360.73 lbs. as shown in Figure 2.11. The total measured deflection at mid-span after the sample was removed from the test apparatus was 7/16 inch. The test of sample 4 was also stopped due to excess deflection at 225.86 lbs. The total measured deflection at mid-span was one-half inch. For the reinforced beam, sample 5 failed when the bond was broken, as shown in Figure 3.12. Note that the crack devel- oped from the top of the beam (tension zone) and gradually increased to the bottom (compression zone). No coolant leakage was observed in the area where the crack developed. Figure 2.13 shows the bond slip within beam sample 5. The slip surface was quite smooth and firm. The beam reinforced with a steel rod with lugs at both ends (sample 6) also failed by bond rupture but the failure mode showed some difference from that of sample 5. In sample 5. the crack was wide open after bond slip occurred. In sample 6. the crack opening was limited by the lug at the end of the beam as shown in Figure 2.14. It appeared that rupture first developed from the top of the beam due to bond failure but displacement of the reinforcement member was limited by the lug. Then. all stresses were transferred to the compression zone which caused a compression failure in .11 ‘h .11 .p. OL. A. LIN 00‘ ...].n. r 2 P. 2 l D.\' 59 the frozen sand followed by development of a crack to the top. The compressive failure in frozen sand at the bottom of the beam can be seen in Figure 2.14. Besides creep response of beams. the initial deflec- tions at each load increment are listed in Tables 2.5 and 2.6. For an average increment of 25.97 lbs., the initial deflection of sample 3 was 0.0007 inch at -10°C, and the initial deflection of sample 4 was 0.0017 inch at -6°C. Since the elastic modulus of tension, Et, is larger than that of compression, EC, for frozen soil, these values can be used to compute the initial tangent modulus if the ratio Et/Ec is known. Tables 2.7 and 2.8 list the initial tangent moduli for tension and compression by assuming different ratios of Et/Ec. Although the true ratio of Et/EC is unknown. the initial tangent moduli listed in Tables 2.7 and 2.8 will be useful for the numerical analysis given in the following chapters. 2.4.2. Discussion Since the beam deflection is a combined effect of creep in both the tension zone and in the compression zone, the steady state creep deflection curves only mean that the stress distribution within the beam has reached a steady state condition. It is not necessary that all creep strain rates within the beam reach the secondary state of creep. Since the creep rate in tension was faster than that in com- pression for frozen soils, the strain in the tension zone 60 may reach the secondary creep state when the strain in the compression zone is still in the primary state. If the load was increasing or the loading time was getting longer, the creep strain in the compression zone will gradually go into the secondary creep state. In Figure 2.9, samples 5 and 6 were both reinforced with a 3/16-inch steel rod but the observed deflection rate of sample 6 was larger than that of sample 5 at the same load. This phenomenon can also be explained as the effect of primary creep. For a reinforced beam part of the tensile stress will be taken by the steel rod. Both tensile strain and compressive strain may be in the primary state when the beam deflection shows a steady state condition. In the primary state, the strain is dependent on the loading time and the previous creep strain. If the loading time or the previous creep strain is small, the creep strain rate at that time will be large. If the loading time or the previous creep strain is large, the creep strain at that time will be small. This effect can be expressed more clearly by the power creep law as ° b n b C- :5. .9. f ‘(b)(oc)t ‘2'" where éc and 0c are the proof strain rate and proof stress respectively. n is the power parameter for stress. b is the power parameter for time. If we adapt a time hardening 61 theory, the creep strain rate can be computed by dec _ (:£>D(J1\ntb-l (2.2) 0 (it b C/ where the creep strain rate dEC is dependent on tb'1. Since at b is less than 1 in the primary creep state, if t is small, c c IE? will be large. When t is increasing,5%%- is decreasing. The same effect occurs if we adapt the strain hardening theory. The creep strain rate can be computed by b-1 . n ' C /b dc - c 5 - 0 dt - (E ) (5C) (T) (2'3) c dec . . . where dt 15 dependent on the prev10us creep strain to a power (b-1)/b. Again. since b is less than 1, if eCiS dec - C ‘ - dec . . small,7fi7 is large. Ife is getting larger. dt 1S getting smaller. In Figure 2.9, sample 5 was loaded at an initial load of 96.10 lbs., and sample 6 was loaded at an initial load of 148.05 lbs. By this time, sample 5, loaded at 148.05 lbs., had already been loaded for thirty-three hours and the previous creep strain was large while sample 6 was just loaded and the previous creep strain was small. Thus, the creep strains in samples 5 and 6 differed by a ratio of b-1 b-1 (cg/cg) 5 (H‘(t5/t6) . This explains why the deflection rate of sample 6 was larger than that of sample 5 at the same load. 62 The deflection of a reinforced beam is a combined effect of creep in tension, compression and bond. The mechanism within the beam is so complicated that it may only be analyzed by numerical methods. Nevertheless, the bi- linear relationship may be explained as a result of the bond creep. When creep rates at the interface are smaller than those in the frozen soil, part of the load is transferred to the steel rod through a bond stress. If the load reaches a critical value where the creep rate on the bond interface is larger than that in the frozen soil, new loads added to the beam may not be transferred to the steel rod. Therefore, the beam behaves as a plain beam. This also means that the steel rod can only take a maximum amount of stress at the critical load. In Figure 2.10, the right-hand side of the bi-linear curve has the same slope with the plain beam. The critical load and the failure load for sample 6 were higher than those of sample 5. This can be explained by the effect of lugs at both ends of the steel rod in sample 6. In the plain beam test, if the shape of the deflection is assumed to be circular the curvature of the beam Y" can be computed by Y" = 2 2 (2.4) whereyfi'is the deflection at the middle span and L is the 63 length between the two supports. The derivation of Equation (2.4) is shown in Figure 2.15. The strain at the extreme fiber of the cross section is Emax = YHZ (2.5) where z is the distance from the neutral axis. Thus, the strain rate in a loading increment can be computed by: t+At _ t Emax Emax (2.6) n o I I max at If we assume the neutral axis is at the middle of the cross section, the strain rate in each load increment can be computed by Equations (2.4). (2.5) and (2.6). The strain rate computed in this manner will be the average strain rate for tension and compression. The average strain rates com- puted from Tables 2.1 and 2.2 are from 1.2x10'7 sec"1 to 3.2x10'9 sec‘1 for -10°c and from 1.0x10'7 sex'1 to 1.5x10'8 sec’1 for -6°C. Note that the strain rate in the plain beam tests are close to the strain rate range of uni- axial tensile and uniaxial compressive tests conducted by Eckardt (1981). a I All s . s .../5 § 11.5 64 Table 2.1. Summary of Test Results for Sample 3 (Plain Frozen Sand Beam at About -10°C). Load Deflection Rates at Temperature (lbs.) Mid-Span (in./hr.) (°C) 75.00 1.231x10‘4 -10.1 101.00 3.625x10'4 -10.3 127.01 6.241x10'4 -1o.3 153.10 1.491x10‘3 -10.3 178.98 1.891x10'3 -10.2 205.01 2.160x10’3 -1o.3 231.01 2.371x10’3 -10.3 257.11 3.545x10‘3 —1o.2 282.95 3.820x10'3 —10.2 308.75 1.015x10‘2 -10.2 334.80 1.330x1o'2 —1o.2 360.73 1.420x10'2 —1o.0 Table 2.2. Summary of Test Results for Sample 4 (Plain Frozen Sand Beam at About -6°C), Load Deflection Rates at Temperature (lbs.) Mid-Span (in./hr.) (°C) 70.07 1.706x10‘3 -5.95 96.10 3.520x10'3 -6.05 122.05 5.601x10'3 -6.05 148.14 6.442x10’3 -6.05 173.99 1.058x10“2 -s.95 199.87 1.155x10'2 -5.95 225.86 1.240x10’2 -6.05 65 Table 2.3. Summary of Test Results for Sample 5 (Frozen Sand Beam Reinforced with a 3/16-inch Smooth Steel Rod at about -10°C). Load Deflection Rates at Temperature (lbs.) Mid-Span (in./hr.) (°C) 96.16 1.901x10'4 -10.05 122.16 2.502x10’4 -10.15 147.90 3.011x10‘4 -10.05 173.88 3.802x10’4 -10.15 225.88 6.601x10'4 -10.10 277.71 1.615x10‘3 -1o.05 329.69 2.221x10'3 -1o.30 381.37 3.180x10'3 .10.30 433.10 5.605x10'3 -10.00 Table 2.4. Summary of Test Results for Sample 6 (Frozen Sand Beam with a 3/16-inch Steel Rod, with a 1/16-inch Lug at Each End at about-40°C). Load Deflection Rates at Temperature (lbs.) Mid-Span (in./hr.) (°C) 148.05 1.041x10'3 -10.00 199.73 1.733x10'3 -1o.05 251.33 1.782x10‘3 -1o.40 303.26 3.463x10'3 -1o.1o 355.18 3.905x10‘3 -10.10 407.15 6.201x10‘3 -10.15 459.06 1.131x10‘2 -10.15 511.05 1.592x10‘2 -1o.15 66 Table 2.5. Summary of Initial Deflections at the Beginning of Each Load Increment for Sample 3 (Plain Frozen Sand Beam, -10°C). Load Load Increment Initial Deflection (lbs.) (lbs.) at Mid-Span (inch) 75.00 101.00 26.00 0.0005 127.01 26.01 0.0008 153.10 26.09 0.0005 178.98 25.88 0.0007 205.01 26.03 0.0010 231.01 26.00 0.0010 257.11 26.10 0.0005 282.95 25.84 0.0015 308.75 25.80 0.0005‘ Average 25.97 0.0007 67 Table 2.6. Summary of Initial Deflections at the Beginning of Each Load Increment for Sample 4 (Plain Frozen Sand Beam, -6°C). I.oad Load Increment Initial Deflection ( l 05.) (lbs.) at Mid-Span (inch) 7'0.07 96.10 26.03 0.0015 122.05 25.95 0.0020 148.14 26.09 0.0015 1 73.99 25.85 0.0015 199.87 25.88 0.0017 225.86 25.99 0.0020 Average 25.97 0.0017 68 Table 2.7. Estimates of initial Tangent Moduli for the Frozen Sand Beams at -10°C. Assumed Et/EC Ratio EC (psi) Et (psi) 1 .80x105 .80x105 2 .40x105 .28x106 3 .48x105 .64x106 4 .96x105 .98x106 5 .60x105 2.30x106 6 .36x105 2.62x106 7 4.18x105 2.92x106 8 4.04x105 3.21x106 9 3.92x105 3.52x1o6 1o 3.82x105 3.82x1o6 f\¢ Ii 69 Table 2.8. Estimates of Initial Tangent Moduli for the Frozen Sand Beams at -6°C. Assumed Et/EC Ratio EC (psi) E (psi) 1 3.86x1o5 3.86x105 2 2.82x105 5.62x105 3 2.40x105 7.20x105 4 2.18x105 8.68x105 5 1.98x105 9.90x105 6 1.91x1o5 1.15x106 7 1.83x1o5 1.28x106 8 1.76x105 1.41x106 9 1.72x105 1.54x106 10 1.67x1o5 1.67x106 .zmfl> Psopm Adv Eopmhm pmop we awhwmfim H.N mpzmflm } minHmS $4". _ bl ammo 1n l H = _ u ._ m n. _ 1 __ 4 :fi pcmaooolllnnrllfi _ . t1- rllcfl pcmaooo mopmfi599ce.ll\ Lx_\).\3\))21 mcmanmz 1I\\ K 11 1 I COAPmHSmcH V\\\V\\\ -\ nopMcOz .HwUHOfl \ \m HIolIHiIIIIIHII-N .QE@.H_ Op... mcmanmz \\\\ poo . mumcuoomm CM 00 "NH." pwmcmm.ll\ p H o :1-Tmmu 09 Wu; \\\V raw a 1/ mvzwwos Ummc 30w hem Empmhm awoficmsomz my wumcwpp pCmEmomemHQ mnwhm I‘||\// 71 Displacement transducer {I‘o Recorders =———- Membrane To Temperature --— - Monitor Hanger ’/ Frozen sandbeam Thermistor Coolant in Coolant out Dead weights Figure 2.1 (continued) (b) Side view. 72 Load transfer bars pso mefip .m> cmomibwm vm coapoma%oo :.m mpsmflm AmpSOLV oEHe ESEEESNHSSSmaomem «Ho . p h p p p p n p L . 00.0 45H mm .Ho.o .u )I|\|.II.I|I-III|\I 65H Hoa 111 1;-...1 .mo.o no 0 \\\|\ To 111111. fimo.o "H mpg Ho.mma m 46.6 m. mcfi od.mmfi m u o.o . . _ M... and mo.mea .\\\\1.wo.o w“ r... _ .mo.c & 2 ”u and Ho.mom .mo 6 an ..M .océ ..1 .oH.o wna HO.HMN m .UMOH smmm rHH.o 0.22 0.21 0.20 .19 0.18 0.17 Deflection at Mid—Span (in.) O 0.16 0.14 0.13 0.12 Figure 75 308.75 lbs 1 L 1 1 ’,,282.95 lbs 257.11 lbs 0 1 2 3 4 5 6 7 8 9 10 117—12 Time (hours) 2.4 (continued) Deflection at mid-span vs. time curves for sample 3. Deflection at Mid-span (in.) Figure .44 ‘ .42 ‘ .41 . .40 4 ~39 ‘ .38« .371 .36 4 334.80 lbs ~35‘ .34. -33‘ .32 .31- .30‘ .29- .28‘ .45 - 360.73 lbs 07 I V V ‘ I v v 1 fl 0 1 2 3 4 5 6 7 8 9 10 11 12 Time (hrs.) 2.4 (continued) Deflection at mid—span vs. time curves for sample 3. Beam load, P 0.18- 148.14 lbs 0.15‘ 0.141 0.12‘ 0.11, 122.05 lbs 0.101 0.091 0.08‘ Deflection at Mid-span (in.) 0.071 0.06‘ 96.10 lbs 0.05“ 0.041 .1 0.03 70.07 lbs 0.024 "’7 0.01‘ 0 1 2 3 4 5 6 7 8 9 10 11 12 Time (hours) Figure 2.5 Deflection at mid-span vs. time curvgs for sample 4 (plain frozen sand beam, -6 C). 0.34 0.33 0.32 0.31 0.30 0.29 0.28 0.27 Deflection at Mid-span (in.) 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 199.87 lbs 4 J . 173.991bs c1 .1 0 1 2 3 4 5 6 7 8 9 10 11 712 Time (hours) Figure 2.5 (continued) Deflection at mid-span vs. time curves for sample 4. 0-55 - 0.54 0.53 0.52 0.51 0.50 0.45 Deflection at Mid-span (in.) 0.44 0.43 . 0.42 0.41 - 0.40 0.39 - 0.38 0-37 0.36 0.49 . 0.48. 0.47 ‘ 0.46 ‘ 79 225.86 lbs V 7 8 9 10 11 Time (hours) fi 12 Figure 2.5 (continued) Deflection at mid-span vs. time curves for sample 4. (I 80 .Apoaa mOH-wOHO madmp cram Cowop% chHQ 90% meH .m> cadmncwe pm opmp Coavooawoa w.m onsmwm Amuc\cflv cmdmucwz pm ovum Cowpooamma OH NIOH m: dIOH dqqqqaq a u —4qqqq< . 4 OH d mademm com: 0 a b 9 OH (SQT) 0901 vocal u a O m 395mm —»b».>~r - p —-.-»- p - moH 81 A0 0H1 .Emmn ccmm Cmsopm omopomcHonv m mHQemm pom mo>nso wEHp .m> mmamivHE pm coHpomHmmO u.m ousmHm AmMSOSV oEHB OH NH MH mm :% MH WH Hm 0H m m n w m 3 m N H o .1 b . . . . . . .1 8.0 6x: 3.8 1 11 . 9: 3.84 a N. 114 8 o 6...: o . 3 1111. an: 86S \\ No.0 1|||II|III.\\ fi 9: $681 no.0 \ m 6...: 2.4.8 4 so om .86 ...... w mDH 00.0Nm . wo.o e . 1. . 3.0m n... 9: Rémm .modm . SSW m. . 0H.omw . 2.0 mnH 0.7mm: . NH.o . m .cmOH ewmm 82 Beam load, P 407.15 lbs 0.15 1 0.14 ‘ 0.13 . 355.18 lbs 0.12-1 / 0.11 . ’//,,/” 0,10“ / 0.09 ./ 0.08- / ”3'26 lbs 0.07 ‘ ,,/’ 0.06‘//////// 251.33 lbs 0.05 ‘/ 0.044 Deflection at Mid-span (in.) ’,,.,’**” 199.73 lbs 0.031 0-02‘ 148.05 lbs 0.013 ‘_’——- 0'00 0 71' 75 3 4 5 6 7 8 9 10 Ile2 Time (hours) Figure 2.8 Deflection at mid-span vs. time curves for sample 6 (reinforced frozen sand beam with lugs at both ends, -10°c). 0 0.24 0.23 0.21 Deflection at mid-span (in.) 0 O {.1 oo 0.17 .27 ‘ .22 - 83 459.06 lbs 0 Y I ' 1'23456789101112 Time (hours) Figure 2.8 (continued) Deflection at mid-span vs. time curves for sample 6. 0.43 . O .42 .41 O .40 ~39 O O O .38 ~37 O Deflection at mid-span (in.) c> ~36 O O .34 ~33 O O .29 O .27 .26 0 ~35 1 84 511.05 lbs § 8‘ 9' 10' 11 I2 Time (hours) Figure 2.8 (continued) Deflection at mid-span vs. time curves for sample 6. .APOHQ m6H1m6Hv 0 OH- 46 msmmp bcmm Cowopm UmopoMchp pom cmOH .m> cmam1vHE pm opmh :onomHywO m.m ohstm Amp£\ch cadminz Pm mpmm COHPomHmoO 10H OH N mun #IOH OH 85 4.44. q . —.4.3«_ H _ ....q.‘ q A H . T l 0 mo - .d was mo.msH n 6864 HaapacH . "m Amuse neon Pm m. mmsH :sz swap coonomcHohv Ema my 6 aHdemm a A p e Mowmmmmw . mnH OH.om u odoH HmeHcH . 86 .Apoag mafi-m6Hv 0 ca. #8 memos Ucmm cmnonm boopo%chn ocm chHQ 90% mmpmp COHPomHmob mo Com.pmasoo 0H.N ohsmHm Ap:\:Hv cmomicHE pm mews COHpomHmoa 1101 NioH «nod oH 4443H 4 1 _13 p; '1 BEAM *5 ppm). LOADIIIEr Figure 2.12 Sample 5 (frozen sand beam reinforced with a smooth bar) failed due to bond slip. 89 Figure 2.13 Bond slip within sample 5 (frozen sand beam reinforced with a smooth bar . 90 Figure 2.14 Rupture of sample 6 (frozen sand beam reinforced with a steel bar with lugs). 91 From AABC R2 - (5L)2 + (R — ym)2 2 2 2 2 2Rym + ym 2 2 2Rym = %L + ym R = it + R 2 2 L + 4ym Figure 2.15 Derivation of equation 2.4. CHAPTER III ANALYSIS OF FROZEN SOIL BEAMS USING ONE-DIMENSIONAL FINITE ELEMENTS 3.1. General In this chapter, an analysis of frozen soil beams using one-dimensional finite elements is presented. The analysis is based on the engineering beam theory. The power creep law and the hyperbolic sine creep law were used to compute the creep strains by an incremental approach. A computer program was written to carry out the calculations. The values of creep parameters obtained by Bragg and Andersland (1980), and Eckardt (1981) were used. Comparisons of numerical results with experimental results are presented. For the analysis the following major assumptions were made: (1) The material is homogeneous and isotropic. (2) A plane section remains plane after deformation. (3) The deformation due to shear is small and negligible. (4) Temperature is uniformly distributed and remains constant . 92 93 3.2. Method of Analysis The beam is first divided into M segments along its length. Each segment is then divided into N layers as shown in Figure 3.1. Each of the M x N elements is assumed to be a uniaxial or truss element. It follows from the assumption that plane sections remain plane, the total strain, cg” , for element m,n is. 8run = eo,m T Ymhmn (3'1) in which, as shown in Figure 3'2(a)"o,m is the strain at some reference axis; qnn is the distance of element m,n, to the reference axis and f; is the curvature of the beam. It is assumed that,within a small time interval at, the stress amn corresponding to 8:0, the elastic strain at the middle of At, remains constant during the interval and is equal to _ e _ T c c a . ' E e ‘ E ( emn ' emn ' fiAemn ) (3'2) . . c . . . in which finn’ eh" and Aemn are the initial tangent modulus, total creep strain and incremental creep strain, respec- tively (Figure 3.2[b]). The incremental creep strain is computed from the creep law as Aemn = fmnAt (3.3) in which finn is the creep strain rate. 94 By equilibrium, ZFX = 0, and EN = 0. The axial force %1 and moment Mm at segment m are N P=i: o A (3.4) n (3.5) where Amn is the area of the element. Substituting Equation (3.1) and (3.2) into (3.4) and . * c 46C , (3.5) and letting Emn = Emn + —7?4 yields N * Pm = i=1 Emn mn (Eo,m + Ymhmn Emn) (3'6) N * Mm = fi=1 Emn hmn Amn ( Eo,m + Ymhmn ' Emn) (3‘7) From Equation (3.6), N N * pm = i=1 Emneo,m Amn + 3:1 Emn(Ymhmn ' emann (3'8) Since eo,m and Ym are independent of y, N N P = e X E A + Y" 2 E A - m o,m ”:1 mn mn m n=1 mn mn mn N * (3.9) 2 ”:1 mn mn mn 95 Then N N * P Y" z E h A z E s A 80 m - m m j=1 mn mn mn n=1 mn mn mn (3.10) N 2 E A ”:1 mn mn Substituting Equation (3.10) into (3.7) and writing N N N * 9ij = EmnhmnAmn’ G = 3:1 gmn’ H = 2:1 Emnhmn’ and K = i=1 EmnemnA'mn yields N P + K m G * :2 g (——+Y"(h --—- -€ ) (3°11) Mn n=1mn H m mn H mn Then N Pm -+ K * M -2 g <——-e 1 Y" _ m n=1 mn H mn (3.12) m I N G where I = 6:1 gmn ( hmn - -fi_ ) After Y" is computed, the stress and strain in each layer can be calculated by Equation (3.1) and (3.2), and the deflection at each segment can also be computed, for example, by the "conjugate beam method". 3.3. Creep Laws As discussed in Section 1.3.2, two creep laws have been proposed to compute the creep strain in dense frozen soil. 96 The power creep law used by Andersland, et a1. (1978) is t (3.13) where éc and Ck are the proof strain rate and proof stress. respectively; n and b are the power parameters for stress and time, respectively. Taking derivative with time of Equation (3.13) yields C . de a b a n b-1 mn _ c mn __Ht—_" ( _TT—) ( ) b t (3.14) Then, for a time interval at, the incremental creep strain C Aemncan be computed as n b-1 b c 8c °mn A = , emn ( b )( 0c ) b t at (315) The preceding equation gives the creep strain increment based on the power creep law and "the time hardening theory". Further, if we rewrite Equation (3.13) as 1. 0 :5 g ~~1 51—"l1b < bf ) (3.16) O C t : c emn and substitute Equation (3.16) into (3.14), we obtain: C damn - c (T) . Omn )% dt emn c “c (3~17) II C) II Ne DC hc 97 Then, the incremental creep strain 08;” for a time interval At, can be computed by (P—gli o S 3 V O1: at (3.18) This increment is based on the power creep law and the "strain hardening theory". The hyperbolic sine creep law used in frozen soil has the form c de 0 mn _ - . mn ——Ef—_ — u Sinh Os (3.19) where 0 and °s are parameters. For a time interval At, the creep strain can be computed by 3.5m = 0 sinh ( dams" ) At (3.20) If we plot equation (3.19) in a semi-logarithmic scale, the curve contains three portions (Figure 3.3) according to Nadai (1938). At very small strain rates (to the left of point A in Figure 3.3) the curve isconcave approaching the horizontal axis asymptotically; for the intermediate rate range (AB), it is nearly an inclined straight line; and for large rates (BC), it is convex. The creep parameters 0 and Cs in Equation (3.19) are equal to (Nadai, 1938) o = o 0 = 20 (3.21) Lu h 1' we 98 where 00 is the slope of the straight line AB; 0 is the strain rate at the intersection of the extension of line AB and the horizontal axis. 3.4. Creep Parameters In order to calculate the mechanical response of the model frozen sand beams described in Chapter II, the creep parameters for the same type of frozen sand must be obtained. Two sets of experimental data, as reported by Bragg (1980) and Eckardt (1981), were available. Table 3.1 shows a comparison of materials used in model beam tests with those in uniaxial tensile and compressive tests con- ducted by Bragg (1980) and Eckardt (1981). Since the materials used in the three sets of tests were quite similar, it is reasonable to use the creep parameters as given by Bragg (1980) and Eckardt (1981) for a finite element analysis of the model frozen sand beams. As mentioned in Chapter II, the strain rates of the model beams tested lay in the strain rate range of Eckardt's experiments. However, the creep equation Eckardt used was not in the same form as Equation (3.13). In order to use his parameters for the latter equation, a conversion is necessary. The equation Eckardt used is m -A k S=et Ow (3.22) 99 where 5 = O1/Oref is dimensionless uniaxial stress; = is uniaxial deformation; e 61/Eref T = t /t is dimensionless time; 8 BW To - T 9 = _____?— is dimensionless temperature; To ' w m0: °o/°ref 1S dimenSionless proof stress, and 01, e1, tB, T are uniaxial stress, uniaxial strain, loading time, and test temperature respectively; rain, Oref’ Eref’ tBW’ Tw are reference stress, reference st reference time and reference temperature respectively; T0 is the melting temperature of ice in any temperature scale and 00 is the proof stress. The paramaters Eckardt obtained for Equation (3.22) are listed in Table 3.2. With a simple rearrangement, Equation (3.22) can be written as 1/ m/A A/m m eref/tref °1 /m A ‘1 ’ x/m (m/Iiiekao (t3) (3.23) which has the same form as Equation (3.13). A comparison of Equation (3.23) with (3.13) gives m/x g = _£____ c tref A k m “c = (I) 9 °0 (3'24) n =1/m bzl/m F‘W S 1 D1 100 Substituting the parameters listed in Table 3.2 into Equation (3.24) yields the creep parameters used for the finite element analysis (Table 3.3). The creep equation Bragg (1980) used has the same form as the power creep law in Equation (3.13) except that the parameter b is assumed to be 1. It means that the creep parameters Bragg reported (Table 3.4) are for the secondary creep state only. Note that the test strain rates in Table 3.4 are higher than that of model beam tests. It appears, however, that the hyperbolic sine creep law may be used to extrapolate Bragg's test data for the lower strain rates of the model beam tests. Therefore, Bragg's test results were replotted in a semi-logarithmic scale as discussed in Section 3.3 (Figures 3.4 and 3.5). The linear segments (for the "compressive“ case, use that representing the lower strain rate) were assumed to correspond to the "almost linear" segment AB in Figure 3.3. The parameters for the hyperbolic sine law were then determined as discussed in Section 3.3 and are listed in Table 3.5. The relationship between the two creep laws may also be seen from Figure 3.6. Note that Figure 3.6 is in log-log scale while Figures 3.4 and 3.5 are semi-log plots. 3.5. Computer Program From the analysis described in Section 3.2, a computer program was written. The main steps in the solution 101 procedure are as follows: (1) (3) (4) (5) At time t = 0, the elastic stresses, Omn' and the elastic mye determined. The stresses, were . e strains, em”, omn' . . C . used to compute the incremental creep strains, Acmn, in a small time increment, at. C mn ,were calculated from The incremental creep strains, A: the creep law (Equations E113] (for the first at), [3.15], [3.18], or [3.20]). Different creep parameters for tension and compressionvere used depending on whether the element wasin tension or compression. The size of at was chosen such that the incremental creep strains, Ae;H,wemaless than 5 percent of the total current strains. Note that this approach (5 percent of the total current strains) was used by Greenbaum and Rubinstein (1968) for creep problems of metals. The total creep strains, 6;" , were computed at the middle of the time increment. The curvature of the beam, Y“, is computed by Equation (3.12). The strain at the reference axis,e[n,vms calculated by T mn’ 0 Equation (3.10), and new total strains, 6 at the middle of this time increment (t+%0t)vmre computed. New stresses, omn’ at time t+%At.“ere computed by Equation (3.2). The set of new stresses, omn,1ms then given an iteration number k andums used to compute, again, the incremental C mn in step (2). Steps (2) to (5) were then repeated and another set of new stresses, 053‘, at creep strains As 102 iteration number k+1 wmsobtained. The two sets of stresses mac required to satisfy the following criterion: k+1 k 1°mn l ' lomnl ' k i 0.05 omnlmax where Pk wasthe maximum stress along the cross- mn max section. If Hus criterion wasnot satisfied, steps (2) to (5) wenaalways repeated to iterate new stresses. k+1 were If the criterion wmssatisfied, the stresses,0mn , assumed to be the desired stresses at time t+%At. With 5 percent tolerance the iteration usually converged in a few cycles. (6) Calculate the deflection of the beam by integrating the beam curvature Y". (7) Let t = t+at and repeat steps (2) to (6) for the next time interval. 3.6. Numerical Results and Discussion In this section, the creep behavior of the test frozen soil beams, as calculated by the computer program, is presented. The numerical results are compared with the experimental results reported in Chapter 11. Also presented are results for different parameters, creep laws and numer- ical methods. Note that the most important difference between these analyses and other creep analyses is that 103 herein different creep properties for tension and for com- pression wemaused. An analysis which used a single set of parameters which are equal to the average of the tension and compression parameters was also carried out for comparison. 3.6.1. Analysis of a Simply Supported Beam by the Power Creep Law A simply supported beam subjected to a pair of constant moments was considered in this case. The creep strain was computed by the power creep law. In order to compare with the experimental results, the end moment was made to equal the load P times the moment arm as shown in Figure 2.4(a). A set of step loads were used as in the experiments. Two temperatures, -10°C and -69C, were considered. For the power creep law, the creep parameters listed in Table 3.3 were used. The elastic moduli in the analysis were obtained so that the (initial) elastic deflections would be equal to the experimental values as listed in Tables 2.6. These moduli are listed in Table 2.7 in Chapter II. Comparisons of numerical results with the experimental results for the -10°C case are given in Figure 3.7 in terms of mid-span deflection. Due to deformations from the test system and the protective membrane around the beam, the initial deflection for the first load in the experimental data contain a seating error. The seating error was adjusted, for the first load only, in such a way that the initial deflection can be comparable to numerical 104 results but the slope of the creep deflection curve, which is more important, remains the same. Comparison of numerical results with experimental results for the -6°C case are shown in Figure 3.8. Both sets of numerical results were computed using the strain hardening theory and an Et/EC ratio of 5. The effects of using different ratios of Et/EC and the time hardening theory will be discussed later. Note that the agreement between the experimental and the finite element analysis results is good for most of the loading levels. If we plot the results in terms of load versus deflection rate in a log-log scale, the comparison appears even better (Figure 3.9). In Figure 3.10 is shown the distribution of stress and of strains across the depth of the beam. It is interesting to note that unlike past creep analyses of beams the neutral axis of the frozen soil beam shifted with time. A transition zone existed in which the tensile strains and stresses would change to compressive ones and vice versa. In the transition zone, singularity would be encountered when the strain hard- ening theory was employed but not with the time hardening theory. In a stress-strain-time plot, Figure 3.11(a), the stress path of the time hardening theory goes vertically (i.e., keeping time constant). A change of sign in stress causes no problem. The stress path in the strain hardening theory goes horizontally in the stress-strain-time plot (i.e., keeping strain constant), Figure 3.11(b). As indi- cated in the figure, the theory precludes any sign changes in 105 stress. For the strain hardening theory, the incremental creep strain, 02;”, is proportional to the total creep strain, 6C , raised to the power of (b-1)/b in Equation (3.18). mn Since 0 is usually less than 1, when the total creep strain approaches zero in Figure 3.11(b), singularity occurs. Therefore, in the analysis using the strain hardening theory, the equation of the time hardening theory (Equation [3.131) was also used to replace the equation of the strain hardening theory (Equation [3.18]) when the total creep strain approached zero. 3.6.2. Analysis of a Simply Supported Beam by the Hyperbolic Sine Creep Law As discussed in Section 3.4, the strain rates involved in the model beam tests lay in the strain rate range of uniaxial tension and compression tests conducted by Eckardt (1981). The creep parameters obtained by Bragg (1980) which are not in the same range as in the model beam tests need to be extrapolated if the power creep law is to be used. There- fore, an analysis which used the hyperbolic sine creep law with creep parameters listed in Table 3.5 was executed. A simply supported frozen sand beam with Et/Ec ratio equal to 5 at -10°C was analyzed. The effect of different ratios of Et/Ec for the hyperbolic sine creep law case will be discus- sed later. The elastic moduli used in the analysis were adjusted to suit the elastic deflection of experimental results. The values turned out to be approximately two-thirds 106 of those listed in Table 2.7. The numerical results compare reasonably well with experimental results on both normal and log-log scale (Figures 3.12 and 3.13). On the other hand, if one uses the power creep law with creep parameters listed in Table 3.4, (data from Bragg, 1980) the comparison is very poor (Figure 3.14). As discussed in Section 3.3, the hyperbolic sine creep law is a curve on a log-log scale plot whileifluepower creep 13" is a straight line. Figure 3.16 shows the curve of hyper- bolic sine creep law comparing with uniaxial compression test data obtained by Bragg (1980) and Eckardt (1981). The curve of hyperbolic sine law bends into the test range of Eckardt's uniaxial compression and tension tests (Figure 3.6) which is also the range of model beam tests in Chapter II. This gives the reason why numerical results compare well with experimen- tal results when the hyperbolic sine creep law was used. The stress and strain distribution along the cross section for the hyperbolic sine creep law is similar to those for the power creep law. Since the hyperbolic sine creep law implies the time hardening theory, there would be no singularity occurring in the transition zone. 3.6.3. Parametric Studies A few cases of changing the solution techniques and varying the creep parameters are considered in this section. 107 Figure 3.15 shows solutions obtained by employing the iteration method (Section 3.5), straight Euler method and modified Euler method (see, for example, Stoer and Bulirsch, 1980). Since the iteration in a time step usually converged in one or two cycles, the solution, based on the modified Euler method, was very close to thosecfi‘the iteration method. Results for the straight Euler method did not com- pare well with that of the iteration method at the early time steps because the system response was changing rapidly and the former method was too crude for its prediction. Figure 3.16 shows a comparison of numerical results using the strain hardening theory and the time hardening theory. Note that the solution was not sensitive to use of either theory (barring the singularity as discussed earlierL Figure 3.17 shows a comparison of numerical results for the power creep law with different ratios of Et/EC. Figure 3.18 shows a comparison of numerical results for the hyperbolic sine creep law with different ratios of Et/EC. Note that beam deflections were more sensitive to the Et/Ec ratio when the hyperbolic sine creep law was used. This may be explained as follows. Regardless of the creep law used, the beam creep behavior should be independent of the modulis ratio, Et/EC, once a steady state stress distri- bution is reached. Generally, this state was reached sooner for the power creep law than for the hyperbolic sine law. This occurred because the effect of primary creep was included in the former and not in the latter. For the power 108 law, the operative time for the ratio was brief and the effect was small. For the hyperbolic sine law, the opera- tive time was larger and the effect was substantial. Figures 3.19, 3.20, 3.21, and 3.22 show, for the power creep law, changing beam deflections with varying creep parameters Oc’éc’ n, andl). The deflection increased with a decrease of 0c and n, and an increase of éc and b, and vice versa. Since ac is the stress denominator of the power creep law in Equation (3.13), when °c decreases, (Zn—C") increases. Therefore, the creep strain and the deflection increase. Similarly, when n decreases cyflgr‘increases (since °mn<°c)a"d the creep strain increasgs. éc is a multiplier in the power creep law; therefore, the creep strain increases whenét increases. The effect of parameter “'0 E b is not so obvious in Equation (3.13). When 0 increases, If) D increases. It appears that the effect of tb b E is larger than (I?) ,and the creep strain increases when b decreases and t increases. Among the creep parameters, n is the most sensi- tive parameter, 0c is the second most sensitive parameter. When computing the results shown in Figures 3.19 to 3.22, the parameters were gradually increased or decreased up to 50 percent. In Figure 3.19, the decrease of 0c stopped at 15 percent because a further decrease of'oC would cause numerical instability. Numerical instability occurred because the creep strain became too large whenaC became closer to %"‘or even smaller than Omn' In Figure 3.21, the 109 decrease of n stopped at 15% because the deflection simply became so large that it could not be plotted in the figure. Numerical results using the average creep parameters for tension and compression were also obtained. Figure 3.23 shows that numerical results obtained in this manner were much smaller than experimental results. When distinct creep parameters for tension and compression were used, the com- puted tensile creep strain was much larger than the compres- sive creep strain. The beam deflection actually was domi- nated by tensile creep strains. Therefore, the deflection computed by average creep parameters was much smaller than that computed by distinct creep parameters. 110 m._ mm.o-m_.o mo-mo mm.o-sm.o No_-ao_ Nassau A_mmlv masts so._ o._-o..o om m.o mow Nptmsc A_mm_v pucmxom m._ mm.o-m_.o 4o mm.o mod Nptaso mpmwp 560m _muoz “as solacoclc: coau:n_tpm_o columtucwucou Amoav to bcaaalttmou bNam caste seam 6106a 6_o> 0:620: sea beam to waxy .AmePV pstaxuw ucm Homo—v ammcm xn now: smock :p_2 mama» smmm “one: cw com: m~owgmuoz Co comwcoaeoo ...m sloop Table 3.2. Creep Parameters Obtained by Eckardt 111 (1981). Compression Tension Reference Values m = 0.62 m = 0.20 0 = 1 KN/cm2 c t ref Ac = 0.16 At = 0.11 tBu = 1 hour ”0c: 210 ”pt: 24 IN = -273°C kc = 1 kt = 1 6ref = 1 Table 3.3. Values of Parameters for the Power Creep Law (Data from Eckardt, 1981). Temperature oc(psi) e(hr’1) n b -6°C Tension 817 5.00 0.55 Compression 8312 1.61 0.26 -10°C Tension 1360 5.00 0.55 Compression 13845 1.61 0.26 112 mic—xmm.. _.om~ cowmmmgqeou mquxom.m m.m~ :oumcmh Ooopa Apiczv a A_mavmo mgauogmaewh .Ampao m.ommcm sous umpafioamtpxmv 360 combo mcwm eagontmaxz map Loy mgwpwEmgoa Co mmzma> .m.m m_nwh so.o_ own ¢-o.xo.~ ow m-o_xm._ o_- :o_mcme mo.m_ Room N13.8.4 op o-o_xm.m o_- co_mmmtaeou :2: 3.; c on A_-ommv spam cgmtpm weaposmaemh .364 ammco cmxoa 0:» to» Hommvv omega »2 noc_muno mcwpmsotma qwmcu .¢.m m~aep 113 .2609 a mo Eopmaw mpouowHO H.m musmHm 1‘21 z r-4 (\1 €'\.:r 114 sea samba .Haupaca do sowsaabm Abs 4 base p<+p ADV p .Hm>hmch oEHp m CH :Hmppm Hapop CE a w w or .H. CEO 0 \ CE CEw B ‘ m. s .s n X41 9 .pmzmH Hmonhp m CH chhpm Amv N.m mpzmHm :wam mocmnowou: 1 / T1 115 1 D U) (D (D S4 .p U) / l / I g, 0 0 log a Figure 3.3 Stress as a function of strain rate for the hyperbolic sine creep law. 116 ieHmm m :0 vmvaHg Aomev mmwnm hp vochon .meom oHscpHmeOH mpHsmoh pmmp m>HmmopQEoo :.m opsmHm AH owmv wvmm chupm ommuo>< S S 1 2 o 0 4‘4 ‘ 3 G 1 .111 1 1 111 1 d .4941! 1 1 d ‘ .11 1 d d d d .4411 4 ‘ 1 1 I L i 1 1. ... I. l .l ‘ A memos o>HmwouQEoo .OOOH: n e 4 T 1 t 1 . a a 4 a . d E) P r hbb‘b h b b P bbbhb b b b b hrbbbL P F b btPPb b OOHH OONH OOmH OOcH OOmH OOOH OONH oom— OOaH OOON OOHN OONN OOmN Oqu OOnN (tsd) sseaqs xead 117 ufiEmm m :0 Umppoaa Aommav wwwum an vmcHMPno “anommv mpmm :fimupm wwmum>< ~-o~ n-o~ euc~ nuod .mamom oasspfipmwoa mpHSmmh 9mm» mHHmCme m.m mpzmflm g-o~ 5-0“ 111‘4 i 1 .111111 1 - .ddddd mpmmp maflmcme .oooa- u a FDFDLF D h h hbr*bp P b P "”’ ’ b > 1 1 ‘ -11dddd d 1 o co“ 4 Joan con ooe oon Head loco " \ 5.! k02 ) ssan noow Ed (I 1com 1000“ Loodd 100mg 1con~ coed L’pbbp >L F 118 .Aawma .pcpwxom can .omoH .mmmpm Eopm «pmuv mpfizmmh pmwv ammpo Hwflxmficz spas o>hzo BMH ammho mcflm oaaoppwgh: map Mo comHHMQEoo w.m myswflm “anomwv wpmm :Hmupw muoa :nOH muoa wuoa muoa wuoa mnoa <4< . ~pFF>L> - - 7.... h . - ——.-_ p. - Lthhuhp_ _ “nOH 119 : 8. O 05 ) -———— Experimental result T’T -—-— Numerical result 3350.0u‘ 9:127 lbs Ev 4‘; 0.03‘ 5 fl 0.02< +3 -_______..._ o q ____-__.-—-----"" #_ 3 OJH ,-——--"’ $4 8 0,00 . v . . v s 1 . . . s 1: O l 2 3 4 5 6 7 8 9 10 ll 12 Time (hours) 2 a A :nr~0.05 ‘ ———— Experimental result .é 5 ~--- Numerical result £30.044 33:101le +3 m 0 03 ‘ S 0H 0.02‘ +3 0 a 0.014 ____________________ ”...,” ‘— 30.00'“',1s.r17v.1 is; O J. 2 3 4 5 6 7 8 9 10 ll 12 Time (hours) g 3 0.05 “ -————-Experimental result 'é»~ ----- Numerical result 1; 0.03d 8 of: 0.02( +3 -.....- O (D H c... Q) Q 0.00 v v ‘ a 1 1 . a 1 s f '4, O .. 2 3 3 5 6 7 8 9 10 ll 12 Time (hours) Figure 3.7 Compsrison of numerical results base on the power creep law with experimgntal results (plain frozen sand beam, -10 C). Deflection at Mid-span Deflection at Mid-span Deflection at Mid-span (in-) CO (in.) 00 (in.) 00 O O O O OO O OO O O O 120 ———— Experimantal result Numerical result 00 1 v T 1 i T 1 V V I; O l 2 3 h 6 7 8 9 10 Time (hours) 05‘) -——- Experimental result --- Numerical result .ou1 P = 173 lbs .03‘ .02 ,/"’ .01 ‘ ””’/ .00 v v w T I 1 I 1 fl 7‘: O l 2 3 h 5 6 ’7 8 9 10 Time (hours) .051) ———— Experimental result --“- Numerical result 044 P = 153 lbs .034 .02‘ .014 .OO ' r r’ :7 8 9 10 Time (hours) Figure 3.7 (continued) Comparison of numerical results based on the power creep law with Experimental results (plain frozen sand beam, -10 C). A 0.07‘ 0.05‘ 0.04‘ .03. 0.02‘ 0.014 Deflection at Mid-span (in.) O 121 -——— Experimental result --—— Numerical result P 2:309 lbs 0.00 0.071? 0.06) 0.05‘ 0.00% 0.03‘ 0.01‘ Deflection at Mid-span (in.) 0.00 6'76'9Y6 Time (hours) ———— EXperimental result ---- Numerical result P = 283 lbs Figure 3.7 7 8 éfo' Time (hours) (continued) Comparison of numerical results based on the power creep law with gxperimental results (plain frozen sand beam, -10 C). Deflection at Mid-span (in.) 0.00 Deflection at Mid-span (in.) O 012345678910 122 —-- Experimental result ———- Numerical result ‘ i 1 1 ' 7 ' ‘ ‘ 1' 012345678610 Time (hours) 4N. -——— Experimental result ‘ --—- Numerical result ‘ P: 231 lbs ,I’ 4‘ I I Time (hours) Figure 3.7 (continued) Comparison of numerical results based on the power creep law with 8xperimental results (plain frozen sand beam, ~10 C). 123 ll ———— Experimental result : 0.05. —-— Numerical result m Q — '{J’TOwa P“ 96 lbs <3: "-1.... EVO.03‘ 13’ c 0.021 O '3 001‘ 8 . H 00 x “-4 ' T Y I v 1 I 1— , ‘ r fir WI 8 0123456789.101112 Time (hours) c a O 05:) ———- Experimental result m —-- Numerical result 'o Egoou‘ = 70 lbs ,_ ..-— -—-— ”2350.034 g ,/ .H 02 / P / O ,3 01‘/ s. Q) g Q 0.00 ' ' ‘ ‘ ‘ ‘ j 1 I7 0 l 2 3 b 5 6 7 8 9 10 ll 12 Time (hours) Figure 3.8 Comparison of numerical results based on the power creep law with experimental results (plain frozen sand beam, - C . III==IIIIIIIIIIIIIIIIIIIIIlIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 124 A\-——-Experimental result 0.08- ----Numerical result ”f _ / C 0.07-i P— 148 lbs /’ :1 / c 0.06« / m o. g? 0.05‘ 'o .H 4 53 0.00 .p ‘“ 0.03« c o I: 0.02‘ O 35 o 01‘ $4 . (D c: 0'000 i 2 3 CE 13 '6 5 8 § 16’ Time (hours) 0.08‘ ——- Experimental result -- Numerical result P = 122 lbs 0.05‘ 0.044 0.03‘ 0.02‘ 0.01‘ Deflection at Mid-span (in.) 0.00 . re . . . . . r w T; 0 1 2 3 U 5 6 7 8 9 10 Time (hours) Figure 3.8 (continued) Comparison of numerical results based on the power creep law with experimental results (plain frozen sand beam, -6 C). 125 h I @mopo pogo .Ao n .5009 dawn Cmuohm cflmaov mpHSmou Hopcosfipomxo spa; 30H m 029 :o comma mvasmos Hmofihossc mo Comanmasoo AcozcflpCoov w.m osswflm Ampsocv oswa m P s h c m z m m b L L mnH OON H & pmsmou Hmofiuoszz pasmop Hmvcoswnooxm 00.0 8.0 No.0 no.0 20.0 m0.0 tioatgaq 00.0 (x O O O\ammoumsoo :flmupm m>fimmmuQEoo Awmmmoauv o H N o Hoo.ou 3000.03 0 m -m .0 d 300.0 0 mmoupm oaamCoa Camuvm oafimcme 128 A“ 0‘ H4 0 Lu P 02 0 U) (compreSSion) 03 ’/——"stress path" ¥ Time 0:. a (tension) (a) L9 01 Stra 02 (compression) 03 . ..-- \-"stress path" ‘ Time— 00 05 (tension) (b) Figure 3.11 (a) Stress path of time hardening theory. (b) Stress path of strain hardening theory. 129 IN g 0-054 -— Experimental result $ --~ Numerical result EZO'OL" ? = 127 lbs ”$50.03* 5 0.021 33 O ‘ ’— ““ 3 0.01 ______________ M ‘ _______ éi 0.00 . . ’fi’ . . r . . . . . , 0 O 1 2 3 u 5 6 7 8 9 10 ll 12 Time (hours) 5, . 3‘ 0.05‘ ———Experimental result ,é --- Numerical result ....‘A 4 5 50'0“ P = 101 lbs ”$30.0? : .3, 0.02. E’ a 0.01‘ __________ g 0.00 WMT 13.—....-.1. — 01'230'5‘6‘00‘91‘01'113: Time (hours) : 0 fr)! 0.05:K —- Ebtperimental result .5 —-- Numerical result £3.00- - P= 75 lbs +>c $30.03' 2 .2 0.02. +> O 3 001+ ___ ________..—————————-- c... Q) 2 0.00 - ‘ 01231456789101‘1112 Time (hours) Figure 3.12 Comparison of numerical results based on the hyperbolic sine creep law with experimental results (plain frozen sand beam, -10 C). 130 Deflection at Mid-span C J a 0.05-) Experimental result ? --—-Numerical result '5’: 1. E0381 P : 205 lbs "-1 4c;~—O.O3. .3 0.024 .p 0 g 0.01‘ c... 80.00........s.> 0 1 2 3 h 5 6 7 8 9 10 Time (hours) A 0.05~-—-— Experimental result -1" Numerical result gO'O“ P = 179 lbs "-1 ~0.O34 \ 0 1 2 3 u 5 6 7 8 9 10 Time (hours) 5 ., o. 0.05«-———-Experimental result 7 —--Numerical result E’TO'OL“ p : 153 lbs +LE 0v0.03‘ 5 ,H 0.024 +9 0 53 0.014 / .8 0.00 ’ . , . - . r r. .T r 22; 0 l 2 3 4 S 6 7 8 9 10 Time (hours) Figure 3.12 (continued) Comparison of numerical results based on the hyperbolic sine creep law with experimental results (plain frozen sand beam, -105). 131 C A 8. 0.05. ————-Experimental result 8 ----Numerical result var-x . . 1 £50.00 P = 257 lbs QVO.O3d /"/’ S x” ..4 0.02‘ l’,’ P ’4’ O J /’ ,3 0.01 /' m / 8 0.00 . T . . , 2 1 1 0 1 2 3 4 5 6 7 8 9 10 Time (hours) C w 8. 0.05‘ ———— Experimental result T -—-- Numerical result :3’T0.044 25 P: 231 lbs tip/0.0} 5 0.02. , s—r” E /”/’ o 0.01* _,, r—I /’ CH 0 000 . 1 . .s 0 ° 0 i 2 3 u 5 0 7 8 9 10 Time (hours) Figure3.12 (continued) Comparison of numerical results based on the hyperbolic sine creep law with expgrimental results (plain frozen sand beam, -lO C . 132 ——~—Experimental result ¢~—---Numerical result ”‘ 07* 0 O P = 309 lbs "-1 ~IO.06‘ C 30 0 5 m ' 5 / ' / E . // E 0.00 / $0.031 / c / .3 0.02‘ / P I o / 0 0.01m r-1 $4 0 D 0.00 Y I y ‘ I i T 0123056285107 Time (hours) \ 0.07. ———— Experimental result —-—— Numerical result 0.064 P’='283 lbs 0.051 0.001 ,1 0.03‘ 0.02- 0.014 Deflection at Mid-span (in.) 3; 53 070’ Time (hours) Figure 3.12 (continued) Comparison of numerical results based on the hyperbolic sine creep law with expgrimental results (plain frozen sand beam, -10 C). 133 .Apoaa moanmoav mmpmh Cowpooamoo ammno opdpm homopm pm mPHSmmp HMpCoEfihmmxm spa; 30H ommpo onwm oHHonquh: on» :o woman mpHSmou Hmowposzc mo somwumasoo mH.m osswflm Asn\cfiv 2000-0“: as spam nonpubfimsa NIOH MIOH JIOH r r Pl D ~.. 2 000m. 4 (1 I 1.- 2—‘I- a . .QKMT. d d i d HOH ‘pBOf I..OH 5 (s01) MOH Deflection at Mid—span (in.) 134 ’11 II 231 lbs T : -10°c Numerical result from hyperbolic sine law \, ’00—’- " Experimental result r—Numerical result from power \\law J__ d I ‘ I T 5 6 7 8 9 Time (hours) - J .1 H N u {—‘1 Figure 3.10 Comparison of numerical results based on the hyperbolic sine creep law and the power creep law With data from Bragg (1980). 135 .39 ‘ s 153 lbs Straight Euler .28, ‘ Method T = -600 .26 1 ./ Modified ,,/” .2“ Euler .I’ Method Iteration Method Deflection at Mid—span (in.) H (3“ .00 “I 1 V V f V Y ' V ‘ fl 0 2 L1 6 8 10 12 111 16' 18 20 Time (hours) Figure 3.15 Comparison of numerical results for different method of solution. 136 .28 ‘ P1: 150 lbs 2h * 22 . Strain Hardening Theory 20 ‘ 18 . Time Hardening Theory Deflection at Mid-span (in.) .00 . . . . w. .2 s - s s 0 2 LL 6 8 10 12 1’4 16 18 20 Time (hours) Figure 3.16 Comparison of numerical results for different hardening theories of the power creep law. 137 .08 - R : moduli ratio NumericalRResuéts (tension/compression) {-R ":2 5 //\R = l 153 lbs Numerical Results Deflection at Mid-span (in.) ..R = 5 ’4‘ _ ,1:;;22?7<“' R - 1 Experimental Result I I I V I “5678 910 Time (hours) Figure 3.17 Comparison of numerical results based on the power creep law with different values moduli ratio. 138 IN ’7 R = moduli ratio (tension/compression) .S .05 . P = 153 lbs v T = -10°c : E3 .04 4 U) 1 3; .03 . 4.: CC C :02 "' . .8 Numerical fl EXperimental Results 5: 0C1 " Result \ __,.—--'" af'rRzl C ’” ___R=5 0 , ‘R=10 Q .00 . . . , - T I Ifi 0 1 2 3 u 5 6 7 Time (hours) Figure 3.18 Comparison of numerical results based on the hyperbolic sine creep law with different values of moduli ratio. 139 .3b . '22 J = : 150 lbs ,x’ '“ T : -6Oc ,”/ 30 “ I" 28 < /" / Decrease o by 15% 26 ‘ I’ 2L}, % I" , / , / / .20 ‘ I / ‘ 8 ‘ ’ With creep parameters from Eckardt (1981) Deflection at Mid-span (in.) ’ ’- 0 2 u 6 8 1012114161820 Time (hours) Figure 3.19 Comparison of numerical results based on the power creep law with creep parameters from Eckardt (1981) and varying parameter 93' 140 .30 P = 150 lbs .28 T : -60C .26 ‘ ,,' ,2u ‘ Increase go by 50%,»“/ .22 . ' .20 . .18 « .16 ~ .1LM With creep parameters from Eckardt (1981) .12 Decrease go by 50% Deflection at Mid-span (in.) .10 .08 .06 .O& .02 .OO 1 j— v 1 fi 1 V v 1 fi 0 2 4 6 8 10 12 lb 16 18 20 Time (hours) Figure 3.20 Comparison of numerical results based on the power creep law with creep parameters from Eckardt (1981) and varying parameter cc. .304 .28‘ .26‘ .2b. .20‘ Q. U .l .16‘ .lb‘ Deflection at Mid-span (in.) .12‘ .10‘ .08‘ .06‘ .04‘ .02d 141 - 150 lbs -600 \ With creep parameters from Eckardt (1981) Increase n by 50% _------ ---———-—---- - -- I"— .OO 8 1012 1416 18 20 Time (hours) Figure 3.21 Comparison of numerical results based on the power creep law with creep parameters from Eckardt (1981) and varying parameter n. 142 ‘n H 150 lbs ,’ .31; ‘ T : -600 x Increase b by 50% ,/ I .J'v' " \ ,l o ,\ \ ,I With Creep Parameters é .28 . /’ from Eckardt (1981) "-4 I, C .26 " ’1’ c: I a 2“ ‘ x ..c x / .Z 022 4 I, I E I ”"’ 4: 20 i I, , ”' a / x’ ,I’ C .18 ‘ I /’”’ C / I 'H‘ .16 . ’ ” +3 I / 8 ,‘ ,’ Decrease b by 10% '—l (H C) :2 .OO . . 4' rv—fi 0 2 1+ 6 8 101214161820 Time (hours) Figure 3.22 Comparison of numerical results based on the power creep law with creep parameters from Eckardt (1981) and varying parameter b. .08‘ O b) Deflection at Mid—span (in.) 'o (\J O H -1 ’ \ "fl" / 108 lb 6 "" {- so - I 143 Experimental gesults P=lh8 lbs, -6 C / I / \ / / / / / / / / ,/ / / / / ,/’ Experimental Average Creep / Result 0 Parameters / P=153 lbs. -10 c 153 lbs, n J C O o C 1 ' V V V 1 123145678010 Time (hours) Figure 3.23 Comparison of numerical results for the power creep law with average creep parameters of tension and compression with experimental results. CHAPTER IV ANALYSIS OF FROZEN SOIL STRUCTURES USING TWO-DIMENSIONAL FINITE ELEMENTS 4.1. General In this chapter. two-dimensional finite elements for plain and reinforced frozen soil are described. The frozen soil was treated as a homogeneous isotropic creep material. The uniaxial stress-strain-time relationship was generalized to a multi-axial stress-strain-time relationship of visco- plasticity employing the von Mise yield criterion and the Prandtl-Ruess flow rule. The creep strain rate in each element was computed by the power creep law with either the time hardening or strain hardening theory. The difficulty arising from the difference in material properties in com- pression and tension was circumvented by weighing the mate- rial properties according to the current principal stresses. To account for the bond relationship between frozen soil and the reinforcing member, two-dimensional bond interface elements were used. The stress-relative displacement rela- tion of the bond interface was assumed to be governed by a creep law similar to the power creep law for the frozen soil. As in the preceding chapter, an incremental method was used for the numerical solutions. Several examples are presented. Numerical results of a plain beam and a 144 145 reinforced beam were obtained and compared with experimental results. Numerical results for a "pull out" test problem were also obtained and compared with experimental results given by Alwahhab (1983). 4.2. Finite Element Formulation For Two-Dimensional Frozen Soil Elements Since the finite element formulation for a homogeneous isotropic creep material has been given elsewhere (Zienkiewicz, 1977; Klein and Jessberger, 1979), only a brief derivation will be presented here for quadrilateral isoparametric elements. The stress-strain relationship for a homogeneous iso- tropic elastic material is 3K-ZG e e Oij = Tekkbij + ZGEiJ- (4.1) E . , E . where K = ——————— IS the bulk modulus, G = ————-— IS the 3(1-2v) 2(1+v) shear modulus; 6ij is the Kronecker delta; and “i3 and e?j are the stress and elastic strain tensor respectively. E is the elastic modulus, and v is Poison's ratio. For a visco-plastic material, the strain tensor eij at any time is assumed to be the sum of the elastic strain tensor efj and the creep strain tensor egj, and we can write 9?. = 3.. - eg- (4'2) 146 Substituting Equation (4.2) into (4.1) yields _ 3K-ZG c c Oij -—3—-—(ekk -ekk)61j+26(eij +613.) (4.3) We assume that the material volume remains constant during creep (von Mise material). It follows that: c .- ekk - 0 (4.4) Thus, Equation (4.3) becomes 3K-ZG .. C Oij - -—3—- Ekkéij + 26(Eij - 8..) (4.5) 1] Equation (4.5) can be written in matrix form as - '1 $111 '1-v v v o 0 0 1’s“ 0 . _ e 22 v 1 v v (l 0 0 22 a J 5. 33 v v 1.v (2 0 0 . 33 ° - E L'_V ‘1 ' - 12 ' (min-m ° ° ° 2 ‘02 ° ‘2 a ‘ ' V Y 23 0 0 0 O. 2 102 23 -v 7 .031. L00 0 O 0 ~2‘_3“ (4.6) - c‘ 6it C e22 ec _E__ 23 1+V VzY12 ‘ c V2723 7ng" 147 For a plane strain problem, Equation (4.6) is reduced to F 1 F1 - l- . F C T 011 -V V V €11 C11 _ E _ E C 4 7 °22 “(1+v)11—2v) " ‘ V1V2 ‘22 ' TE ‘ 227 ( - ) .. v ‘Y C L012; .0 O 2 J .. 12.. ' 1.3/z.Y 12.. . For a plane stress problem, Equation (4.6) is reduced to - ' - - ' P 1 c v c- °11T ‘ V 0 211 E (W) ‘11 T (W) ‘22 E _ 1 c v c “22 = 2 V 1 0 t:22 W ‘17-'17) ‘22 + (T37) ‘11 (4'8) 0 (1-v ) 0 1'V '7 C ‘2.) 0 T ‘2 L ("12 For an axisymmetric problem, Equation (4.6) can be written in cylindrical coordinates (r,z) as - P 1 For. T 1-1_v v v 0 Er q a: C “2 v 1-v v 0 s2 ‘2 = E - E C (4.9) 06 +V _ V) V V 1-V 0 66 W 86 1-2v c orz 0 0 0 —2— er [IIYT‘Z 1- .. L .1 .. u . Computation of the creep strains will be discussed subse- quently. I Consider a quadrilateral element as shown in Figure 4.1. The displacement functions (u,v) in the (x,y) plane (or in the [r,z] plane for axisymmetric problems) are related to nodal displacements {q} by the interpolation functions, [4],: U = [4] {q} (4.10) V where {q} = [91. V1.u2.v2,u3.v3,u4,v41 148 For a bi-linear interpolation, N 0 N O N 0 N 0 4 0 N1 0 N2 0 N3 0 N (1+E)(1-n)/4 (1-E)(1+n)/4 where N1 (1-§)(1-n)/4, N2 N (1+£)(1+n)/4, N 3 4 The strains{e} are related to nodal displacements as: {e} = [B]{q} (4.11) where [B] is the displacement-strain transformation matrix. The potential energy U in an element is: u = blvlelTEDHeldv - 261114115)“ WW“ 14 12) = v.1v1 q}T[B]T[DJ[BJ{q lav - 261,,{qflt31712‘1dv - {q lTlpl in which [0] is the elasticity matrix and {p} is the nodal load vector. The element stiffness matrix can be determined from the theorem of Stationary Potential Energy as: 6U = O (4.13.) Substituting Equation (4.11) and (4.12) into (4.13) yields: 6U = fv[B]T[D][B]dv{ q }- Zva[B]T{s}dv - {p} = o (4.14) which can be written as: 1k.11<11= {2149?} 14.15) 149 where [kf] = fv[B]T[D][B]dv = the element stiffness matrix for frozen soil; {p%} = 26fv[B]T{sC}dv = the pseudo-force vector. 4.3. Computation of Multi-Axial Creep Strains in Frozen Soil As mentioned previously, the creep strains in frozen soil are computed in accordance with the incremental theory of plasticity (with the plastic strain taken as the creep strain). The Prandtl-Ruess equation for plastic strain increment is. P - deij - Sijdx (4.16) in which Si' is the deviator stress tensor and dxis a scalar. J For a Mises material the yield condition is : _ 2 - J2 - VasijSij — K - a constant (4.17) in which J2 is the second deviatoric stress invariant. Substituting Equation (4.16) into Equation (4.17), we obtain 2 _ p p 2 (d1) - tdeijdeij/K (4.18) Using Equation (4.18), Equation (4.16) may be written as dsP. = s.. (4.19) 150 - - 0:2 p 012. ° in which dse _ (3 dsijdeij) - the incremental effective plastic strain"; er~l3J2 = the "effective stress". Equation (4.19) is now rewritten for incremental creep strain, or creep strain rate as o 6 (DO 6e - 3 ij - 2' a Sij (4.20) (D -C in which 2 " " )V' - (2 e ‘ I‘ij‘ij In a uniaxial test the effective stress and the effective = effective creep strain rate. creep strain rate are equivalent to the axial stress and creep strain rate, respectively, i.e. (4.21) 0 II 0 fl 0) (D II No ‘0 Guided by this fact, we postulate that the relation between “e and E: in the uniaxial case also holds for a multi-axial stress state. If a power creep law is adopted, the effective creep strain rate can be computed by E O n - g: = (-59)b(-q—e-)tb 1 (time hardening) (4.22) c 0‘ b-1 a n éc s (cc) 5 6 (—3)5 (strain hardening) (4.23) e e C ac where ac, éc’ n, and b are creep parameters obtained from uniaxial tests as described previously. Since frozen soil has two sets of creep parameters, one in tension and another in compression, a problem arises when 151 it is subjected to multi-axial stresses involving both tensile and compressive stresses. Neither creep parameters in tension nor in compression can be applied. To overcome this problem, a set of effective creep parameters has to be interpolated from those for tension and compression. Since the effective creep strain rate is dependent on the effective stress in Equations (4.22) and (4.23) and the effective stress is a function of the principal stresses, it is reasonable to assume that the effective creep parameters are also dependent on the principal stresses. Since there is no analytical equation available, a linear interpolation of the effective creep parameters was first considered, i.e. Zla-la e _ 1 c ‘c ’ 2104 ‘ ne = 51311:: (4 25) Zloil zlolb i b“ (4.27) Zlafl winiac = 0:, éc = £2, n 2 nt, b = bt if ai is tensile; c . -c _ c _ c . . . °c = °c’ CC = Ec’ n - n , b - b if Oi lS compre551ve. where (0:, s: nt, bt) and (a: s: nc, be) are the tensile and compressive creep parameters, respectively. 152 Equations (4.24) to (4.27) have the following characteristics: (1) (2) If all three principal stresses are tensile (or com- pressive), the effective creep parameters are equal to the tensile (or compressive) creep parameters. For a material in two different stress states, if the principal stress in tension (or compression) is larger than that in compression (or tension), the effective creep parameters are closer to the tensil (or compres- sive) creep parameters for that stress state. These characteristics can be illustrated by the following examples. Example 1: Assume 01:300. 02:200. 03:100. n‘=1.s, nt=4.5 then, ne = [3001)14.5+j200jx4.5+|100jx4.5 = 4 5 [300|+lzoo]+|100| ° where 4.5 is equal to nt. Example 2: Assume nc=1.5, nt=4.5 if, as case A, 01:200, 02='100, 03:0, then, 1 |2001+|-100| ' If, as case B, 01:100. 02=‘100, 03:0 then, ne : 1100jx4.5+j-100|x1.5 = 3 O 2 [1001+l-100] ' Note that n? is closer to nt than n3. Unfortunately, the numerical results using thesevunght- ing functions did not compare well with experimental results 153 and the one-dimensional numerical results in Chapter III as shown in Figure 4.6.* The creep strains computed using the creep parameters interpolated by Equations (4.24) to (4.27) were too small. In order to increase the computed creep strain, the effect of the principal stress on the effective creep parameters must be increased further. Therefore, Equations (4.24) to (4.27) were modified as 2( 0i Pa ! a: = —2|:-|—ai— C , P = 02/02 (4.28) q. . 2 a. e . . a: : (1];IIQC)’ q = ec/e: (4.29) he - £(lalirn ), r = nt/nC (4.30) S be 2( 0‘ 2 ), s = bt/bc (4.31) Note that the parameters p, q, r, s are all greater than 1. Equations (4.28) to (4.31) simply magnify the influence of- the principal stresses. It can be illustrated as follows. *Figure 4.6 will be discussed further in Section 4.6.1. Example 3: 154 Assume nc, nt, 01, 02, 03, are the same as case A in Example 2. With Equations (4.28) to (4.31) ne : 1200|3x4.5+1-1oo|3x1.5 = 4.17 3 |2oo|3+|.1oo|3 e t where n3 is much closer to n than nfi. Besides these two characteristics in Equations (4.24) to (4.27), Equations (4.28) to (4.31) have one additional characteristic as follows: (3) For the same stress state, if one of the creep parameter ratios (p, or q, or r, or s) for one material is larger than that of another material and the other creep parameter ratios are the same, then the effective creep parameters for the former material will be closer to the tensile (or compressive) creep parameter if the princi- pal stress in tension (or compression) is larger than that in compression (or tension). This can be seen from Example 4. Example 4: Assume 01:200. 02=-100, 03:0 if n‘=1.5, nt=4.5 then n§=4.17 as in Example 3. The ratio of ng/nt=4.17/4.5=0.927. If n‘=1.5, nt=6.0 then: ne _ Izoo|4x6.0+1—1ool4x1.5 - = 5.74 4 |200|4+|-1oo|4 The ratio of nf/nt=5.74/6.o 0.955. Since 0.955 is greater than 0.927, n: is closer to t e n than n3. 155 Besides the creep parameters, the elastic moduli and Poison's ratios for tension and compression are not always the same for frozen soil. To evaluate the equivalent elastic modulus and poison's ratio for use, the same form of weighting function was applied. a 2 a. E 15‘ = I‘la , a = Et/Ec (4.32) 210.] 1 c 2 a v ve = l llc , C = vt/vC (4.33) 2|a.| 1 where E = Et, v = vt when ai is in tension; E = EC, v = vC when ”i is in compression. 4.4. Material Model for the Bond Interface As discussed in Chapter II, the bond behavior of a struc- tural member in frozen soil is so complicated that it can only be analyzed by numerical methods. In order to use the finite element method, the joint element introduced by Goodman, et al. (1968) and Ghaboussi, et al. (1973) was modified to model the bond behavior. This element, named the bond inter- face element, had two characteristics: (1) the constitutive relationship in the direction parallel to the bond inter- face was assumed to be viscoplastic (creep); (2) no thickness was assumed in the direction normal to the bond interface. The constitutive equation of the bond interface and a brief derivation of the element stiffness matrix are given asfollows. 156 Since we assume no thickness in bond interface, the analysis of bond behavior works on the stress-relative displacement relationship rather than the stress-strain relationship. For the bond interface between the structural member and frozen soil the following stress-relative dis- placement relationships were used. For the stress normal to the bond interface (4.34) where °n and 6" are the stress and relative displacement normal to the bond interface respectively. 5nf is the relative displacement normal to the bond interface at failure. The relationships are shown in Figure 4.2(a). For the stress parallel to the bond interface, _ t c . as - C$(5S - as)1f (’ssbsf _ . (4.35) °s'0 1f 155>155f t where °s and 6s parallel to the bond interface respectively. 65f is the are the stress and relative displacement maximum relative displacement parallel to the bond interface at failure. a: is the relative creep displacement parallel to the bond interface. The inter-relationship among the stress (total) rela- tive displacement, creep relative displacement and time may be seen from Figure 4.2[b]. At time t, the stress parallel 157 to the bond interface is o: and the corresponding total relative displacement is 6: (Ed in Figure 4.2[b]. which . . . . C contains an elastic portion 6: (EB) and a creep portion 65 t (53). From trinagle abc the stress as is equal to a con- stant CS times 6:. Since 35 is equal to 33-53, a: can be written asaE-o: and Equation (4.35) is obtained. When time changes from t to t+At, the creep relative displacement a: (53 in Figure 4.2[b]) changes to (5:)t+At (s'a' in Figure 4.2[b]) and the total relative displacement 6: changes to aETAt (ad'). Then, from triangle ab'c' the corresponding stress aETAt can be determined. Note that t+At S from the equilibrium equation for the structural system. (6:)t+At is obtained is computed by the creep law and 5 The stress-relative displacement in Equations (4.34) and (4.35) can be expressed in matrix form as %1 Cn 0 an, 0 = - C (4.36) S C % 0 Cs 65 65 Equation (4.36) describes the constitutive relationship of the bond interface. For the bond interface between two quadrilateral elements, 6 and 6S represent relative displace- n ment between two elements. The positive and negative signs of as denote the slip directions as indicated in Figure 4.3(a). The positive and negative signs of on mean that two elements are separated or overlapped (Figure 4.3[b]). 158 The relative creep displacement along the bond inter- face can be computed by a power relationship as (4.37) where 5c and a are the proof relative displacement rate sc and the proof shear stress respectively; n and b are the power parameters for stress and time. Figures4.4(a) and (b) show the bond interface element used in the finite element analysis. The nodal displace- ments at 1,2,3,4, are transformed to the relative displace- ments at i,j as (Figure 4.4[a]) u1 v1 6x1 1 o o o o 0.1 o u2 = 22.12-1131 :2 xj 3 .‘yj. o o o 1 0-1 0 0 v3 u4 .V4 Then, the relative displacements at i,j are transferred to the relative displacements in directions normal and parallel to the bond interface (Figure 4.4[b]). bni -s c O O bxi bsi c s O 0 byi 5nj = 0 0 -s c bxj (4.39) ésj O O c s 6y] where s = sin e, c = cose. 159 For a bond interface element, the displacement field can be interpolated by the shape function 6hi 6 N. 0 N. O a . " - [I J ] "J (4.40) 65 0 Ni 0 Nj bsi bsj where: Ni = 1(1 - E) Nj = 1(1 + E) c -25. Combining Equations (4.38), (4.39), and (4.40) yields W 6 IIJJJJ cNi sNi cNJ. ij -ch -sNJ. -cNi -sNi [~sN. cN. -sN. cN. sN. -cN. sNi -ch] (4.41) 0'” {6} {"3111} With Equations (4.31) and (4.36), the potential energy of the bond interface element can be computed as ” = 1.1411111111611111“ dA - C4,14 1‘1111‘14‘1 dA -1q 1T1?) (4.42) 160 With the theorem of Stationary Potential Energy, we have 6U = (AtmlTlclEMJ dA{q}-CSJA[M]T{6C} dA - {P} = o (4.43) or 11,114 1 = 1P1+1P§1 (4.441 where [kbj = JA[M]T[C][M] dA is the element stiffness matrix; {DE}'= CSIA[M]T{5C} dA is the element pseudo-force vector. 4.5. Computer Program The element stiffness matrice in Equations (4.15) and (4.44) have the same forms and can be assembled to form the structural stiffness matrix by the direct stiffness method as [Ki (4 1 = 1P1+1P‘1 (4.451 where [K] = z[kf]+z[kb] is the structural stiffness matrix; {q } is the global nodal displacement matrix; {P} is the applied force matrix; {PC}=Z{ Pf: }+2{ PE} is the pseudo-force matrix. With the incremental method, the solution procedures of the computer are as follows. 161 (1) At time t = O, the elastic stresses are computed and are assumed to remain constant during a small time interval, at. (2) The creep strains in the frozen soil elements are com- puted by the creep law with the creep parameters weighted by the weighting functions in Equations (4.28) to (4.31). The relative creep displacements in bond elements are computed by the creep law in Equation (4.37). (3) The creep strains in the frozen soil element and the relative displacements in the bond element, then, are compared with the previous strains and the previous relative displacements to make sure that the differ- ences are less than prescribed tolerances. If not, the time increment is decreased and the creep strains and relative creep displacements are recomputed. (4) The creep strains and relative creep displacements are integrated to form the pseudo-force vector in Equations (4.15) and (4.44). (5) The pseudo-force, then, is added to the load vector to solve for the displacements at t+At. New stresses and strains, then, are computed for the next time increment. (6) If the effects of geometry changes are to be considered, the nodal coordinates would be updated by adding the displacements to the original coordinates and the structural stiffness matrix reformed. 162 (7) Steps (2) through (6) are repeated as necessary. For the solution procedure described above, a finite element program was written. The program uses the so-called "dynamic allocation“ of memory to economize computer memory usage. Dynamic allocation means that the needed computer memory is stored in a COMMON block. Entry of every array is kept track of by different pointers. It allows computer memory to be used more flexibly and efficiently. The data input technique which includes the generation of missing nodes and elements and the assembly of structural stiffness matrix from element stiffness matrix follows NONSAP (A Structural Analysis Program for Static and Dynamic Response of Nonlinear Systems). The structural stiffness matrix is stored in a one-dimensional array. The SKYLINE storage technique in NONSAP is used to store the structural stiffness matrix. Subroutines SKYFAC and SKYSOL, developed by Felippa (1975), are used to triangulize the structural stiffness matrix and solve the system of equations. The program is so written that it allows prescribed displacement input. Mixed force and displacement inputs are also permis- sible. Since a large amount of computing time was needed to perform the step-by-step calculations, the program was pre- pared in such a way that it allowed users to stop at an intermediate time step to check the solution status and restart again. Tapes were used to store all relevant infor- mation in the computer at the end of the last time step for 163 restarting the computation. Changing input data at the intermediate time step was also possible. The computer pro- gram itself is listed in Appendix C. Choice of different hardening theories, updating coordinates, refinement of time step, etc., can be done by changing input data. 4.6. Numerical Results and Discussion Numerical examples of using the frozen soil element and the bond interface element developed in the previous section are presented. A plain frozen sand beam and a reinforced frozen sand beam were analyzed to compare with the experi- mental results in Chapter II. A frozen soil cylinder with a steel bar embedded in the center was analyzed to compare with the "pull out“ test results obtained by Alwahhab (1983). Two kinds of "pull out" tests were analyzed: a constant load and a prescribed displacement rate. 4.6.1. Analysis of a Plain Frozen Sand Beam The model frozen sand beam number 4 in Chapter II was analyzed using quadrilateral isoparametric elements, and the weighting functions in Equations (4.28) to (4.31) were applied. The creep parameters in Table 3.3 (Chapter III) were used for computing the creep strains. The elastic mod- uli corresponded to values for the Et/EC ratio equal to 2, as listed in Table 2.7. The effect of the Et/EC ratio will be discussed later. Since the frozen sand beam was loaded symmetrically, 164 Figure 4.5 was used. Numerical results compared well with experimental results (except for the first hour) and with that of the one-dimensional analysis in Chapter III. As in Chapter III, the seating error of the experimental results was adjusted. In the first hour, the numerical results did not compare well with the experimental data due to two rea- sons. First, as the time interval size used was equal to 1 hour, the deflection within the first hour was not known. It may not vary linearly from t=0 to tal hour, as indicated in Figure 4.5. Second, due to the seating error, the ini- tial phase of experimental results was not accurate. Stress distribution along the cross section was similar to that of a one-dimensional analysis. The neutral axis moved up or down as in one-dimensional analysis. The beam deflection was not sensitive to either the time hardening theory or the strain hardening theory as in one-dimensional analysis. Unlike the one-dimensional analysis, singularity did not occur when the strain hardening theory was used. Since the two-dimensional elements were simultaneously sub- jected to stresses in both directions, changing of stresses did not follow the stress path as in one-dimensional analy- sis. The change of stresses was through the effective stress and the effective strain rate which do not have a positive or negative sign. Therefore, no singularity occurred. The Et/EC effect ratio was also studied. In the analy- sis, Equation (4.32) and in iterative procedure were 165 t and EC employed to calculate the initial deflection if E were not identical. The iterative procedure started with a linear analysis using the average value of Et and Ec for E9. After the principal stress tensor was computed for each element, Ee was calculated from Equation (4.32). The new value of Ee (in general, different for different elements) was used for the next linear analysis. Convergence occurred when the principal stresses were sufficiently close. When the Et/Ec ratio was large, it took more cycles to converge. The effect of the Et/Ec ratio on the initial response was also dependent on the grid. For a rough grid, when a large Et/Ec ratio was used, the initial deflection of the beam would be smaller. For the grid shown in Figure 4.5, the effect was small. The Et/Ec ratio had little effect on the beam creep deflection, as in oneedimensional analysis (Chapter III). Due to high computing costs for the analy- sis, studies on the effect of the Et/EC ratio were based on three time steps only. The largest aspect ratio of rectangular elements in Figure 4.5 was 1/2.67. The initial deflection at midspan in Figure 4.6 was 8.97 x 10'3 inch. The value based on the 3 inch, with a usual engineering beam theory was 9.37 x 10" difference of about 4 percent. It was assumed that the numerical results obtained were reasonable. 166 4.6.2. Analysis of a Reinforced Frozen Sand Beam An analysis of a reinforced frozen sand beam (model beam number 5 in Chapter II) is presented. The grid shown in Figure 4.7 contains steel elements, frozen soil elements, and bond interface elements. The frozen soil elements have the same material properties as the previous example. The creep parameters in Table 3.3 (Chapter III) were used. To simplify the analysis, the ratio Et/Ec was taken as 1. Since the effect of Et/Ec was small in the previous example (a plain frozen soil beam), it is reasonable to assume that the effect will also be small in this example. To analyze the reinforced frozen sand beam as a two- dimensional problem, the real cross section was idealized as shown in Figure 4.7. A layer of idealized steel elements was used. Thickness of the idealized steel element was equal to the diameter of the reinforced bar. The elastic modulus of the idealized steel element was computed as shown in Figure 4.7. No creep was assumed in the steel element layer, which meant that creep within the reinforcing bar was neglected. A bond interface element was placed between the steel element and the frozen soil element. For the bond interface elements, creep parameters obtained by Alwahhab (1983) were used (Figure 4.8). Note that the data in Figure 4.8 are for steady-state creep only; thus, the parameter b is equal to 1. The initial bond modulus CS for the bond interface was also estimated from results of the "pull out" tests obtained by 167 Alwahhab (1983) (Table 4.1). The initial bond modulus Cn was assumed equal to CS. Again, only the right half of the beam was analyzed due to symmetry. A comparison of numerical results with experimental results is shown in Figure 4.9. Note that the seating error for experimental results was adjusted as in Chapter III. The numerical results and experimental results did not com- pare well in the first time step for the same reasons dis- cussed in the previous example. Figure 4.10 shows the bond stress along the steel bar on the top and at the bottom layers of the bond interface elements. In the middle portion of the beam, the bond stress was very close to zero, since the beam was subjected to pure bending in that portion. In the beam end portions, the bond stress increased as time elapsed until a steady— state condition was reached. Figure 4.11(a) shows the tensile stress in the steel elements. The stress was constant in the middle portion of the beam along the steel bar and gradually decreased to the end. As time elapsed, the tensile stress increased until a steady-state condition was reached. Figure 4.12 shows stress distribution in the x-direction in the frozen soil at the cross section near the midspan of the beam. As time elapsed, the area of tensil stress decreased. Combining Figures 4.10, 4.11(a), and 4.12, note that the tensile stress in the steel bar and the bond stress in the bond interface increased, while the tensile stress in 168 the frozen soil decreased as time elapsed. This means that the tensile stress in frozen soil was transferred through the bond interface to the reinforced bar when creep in frozen soil occurred. The tensile force and the bond stress in the reinforced bar at the steady-state condition (t 10 hours) are shown in Figure 4.11(b). 4.6.3. Analysis of the "Pull Out“ Test with a Constant Load The constant load “pull out" tests conducted by Alwahhab (1983) were analyzed. Figure 4.13 (upper right- hand corner) shows a smooth steel bar embedded in a cylinder with the bar pulled by a constant load P. Since this is an axisymmetric problem, axisymmetric elements were used. Since the material used in the "pull out" tests was the same as that in the model beam tests, the creep parameters in Table 3.3 (Chapter III) were used for the frozen soil ele- ments. For the bond interface elements, creep parameters and the initial bond moduli in Table 4.1 were used. The elastic modulus and Poison's ratio were taken as 3 x 107 psi and 0.25 for the steel bar. The initial tangent modulus in Table 2.6, with the ratio Et/Ec equal to 1, was used for the frozen soil. Figure 4.14 shows a comparison based on numerical results with experimental results for the "pull out" test. Since the displacement measured in the "pull out" test was between the top of the steel bar and the bottom, as shown in the upper right-hand corner in Figure 4.13, the displacement 169 shown in Figure 4.14 was the displacement at the top of the steel bar (the bottom was fixed). The comparison did not agree well in the first two hours because the input creep parameter for the bond interface included only the steady- state creep while the experimental results represented pri- mary creep in the first two hours. Figure 4.15 (a) and (b) show bond stresses along the steel bar. The stresses parallel to the bond interface, initially distributed nonlinearly along the steel bar, gradually became uniform as time elapsed. The stresses nor- mal to the bond interface were very small and did not change much with time. Figures 4.16, 4.17, and 4.18 show the stress distribution in the frozen soil cylinder at zero, ten and thirty hours. High compressive stresses were concen- trated in the area near the bottom of the steel bar. High tensile stresses were found in the area near the top of the steel bar. As time elapsed, the maximum compressive stress decreased and the maximum tensile stress increased. Figures 4.16, 4.17, and 4.18 also show that the tensile stress in the minor principal direction increased with time. Note that from ten to thirty hours the stress changes were very minor. 4.6.4. Analysis of the "Pull Out" Test with a Prescribed Displacement Rate A constant displacement rate ”pull out" test con- ducted by Alwahhab (1983) was analyzed. The grid in Figure 4.13 was used but the load P was replaced by a prescribed 170 displacement. The input displacement rate was 6.967 x 10'4 in/min. The interval for each time step was one minute. The same material properties as in section 4.6.3 were used. Figure 4.19 shows a comparison of numerical results and experimental results for this case. Since the bond creep model used did not include tertiary creep, the displacement computed did not drop as in the experiment. Figure 4.20 shows the shear stress distribution on the bond interface. The stresses increased with displacement input and gradually became more uniformly distributed. Again, stresses normal to the bond interface were very small compared with stresses parallel to the bond interface. Figures 4.21, 4.22, and 4.23 show the stress distribu- tion in the frozen soil cylinder at zero, five. and twenty minutes. Again, note that high compressive and tensile stresses concentrated in the areas near the bottom and top of the steel bar. The stresses increased with the displace- ment output. From zero to five minutes, the stresses increased faster than from five to twenty minutes. It seems that the stress should approach a steady-state condition as the load in Figure 4.19 gradually be became constant. 171 Table 4.1. Estimate of the Initial Bond Moduli from Experimental Results of the "Pull Out" Tests Conducted by Alwahhab (1983). Temp. No. Initial Initial Area Initial Bond Loading Displacement . 2 ModulusCS (°C) (lbs.) (in.) 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VV/ V I V Vm/L V V??? o- IJ. _ \.\ - _ OW! / 101 .01 IQI 1‘ * t * all . OM1z1 .. \ \\\. ,, / / \\\W\\\u * ,OHU/xll 11 .11 \x :\_\ _ ’ x A x : Sm .11 \\ O\\V\\__\ ’ / / / \ ... n 1. .\x \; ”x\ w x / \f \\ —. .mH \\ / 1. }\ \J x\\ I / / "h 1 . , AND . . pan .\ \ \~ \ 1.1 Hmwpm ’ / /1 _. /1 11Hmmpm m. 3- S _S\ \1 / . / ccmm \ \\w \ 9.8m / / x. CQNOHGHI.‘ \ cmNOpHml . \ a 1%. x / max cl” 0 fi- 11 ‘\ V\o~/flH I / Nm \‘fmm 190 A Prescribed Dis lacement 2 O . Rate: 6.97x10- in/min T = -6°c {a 1.5‘ 01-1 x: 'o m .3 1'0 " 2-D F.E.M. Solution 0.5“ \\ Experimental Result (test S-3, Alwahhab, 1983) 000 I I 1 I T: O 5 10 15 20 Displacement (lo-Bin.) 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I I/ / _// \ \\ 0m / OMI/x 191 1‘ O\\\\ v.\ I / / / / \ : 71mm / .1 \\ LI v. 7, / .i - / \ \I z / IOFI\ \ \ Om / I \ .1\ \ x. .4 \. pm: I /,/ I ./ Lap \ \ 1r1 Hoopm // . .mm 11.Iompm .1 \ Q 4 \ I 1 x, 7. \or / 0L, 1% 02mm \\ “I. \\ Ucwm I / z/ / I I,/ :oNopm111 _ \ comopm.111 -1 , , ... \\ _.\ \ I /S 1--.,1/1/ \ _ -\\- ... \ .\ / I / xo ms 194 .opmp PCmEoomHamHU ownflpommpq m Spa; pmmp :pso Hana: one pow .CHE om u 9 pm Hfiom Cowogg CM Coawzpflppmflc mmoppm mm.: opzmflm mCOHpomth Hmmflocfipm can mCOfipompfiQ Hmaflocflpm wcm Iflmav mmmmmgpm deflocflhd yoga: Iflmav mommmhpm Hwaflocfihm poflmz IwwwwrrIrI/IIIVVIVIH IrvIwwwIervkLVI/H m H- I mma1 .1 1. 1. I. I , I I \\\H I I // \\\ 11 11 .1QG W. _ I I// / LV I I Ellen I on» / \\ m5 9 _ / \ / -1, _ 1. A\ \\I ox. I @m I 1,. 1, \\\ m\ I pan / I _ amp ... .3 1 \ 1128..“ I/ I If x 1 3.: UCMm 11 \\ ow \. \. .Ucmm I ///, 7./ / Cmmop .11 \ Cowopm111 I m .1 \ \.M\ .\ “u x / /L(KWMW \ \ /.. m CHAPTER V SUMMARY AND CONCLUSIONS 5.1. Summary Experimental Work: Two plain frozen sand beams were tested at -10°C and -6°C. A standard silica sand with a specific gravity of 2.65 and a coefficient of uniformity of 1.5 was used to prepare the samples. A sand volume concen- tration of 64 percent was chosen. The beams were loaded in pure bending by successive step loads. Experimental results showed that the deflection rate and load had a linear rela- tionship in a log-log scale. The deflection curve of the beams showed both primary and secondary creep stages. Two reinforced frozen sand beams were also tested at -10°C. One beam was reinforced with a smooth steel bar and the other with a steel bar with lugs at both ends. Experi- mental results for both reinforced beams showed a bilinear relationship between the deflection rate and loading in a log-log scale. The beams first deformed at a slower deflec- tion rate for smaller loads, then changed to a faster rate after a "critical load” was reached. For loads larger than the critical load, the relationship of the deflection rate and loading in a log-log scale for reinforced beams was similar to that of plain beams. For the beam reinforced with a steel bar and lugs, the critical load was higher than that of the beam reinforced with only a smooth steel bar. 195 196 One-Dimensional Finite Element Analysis: For purposes of analysis, a plain beam was divided into a number of layers through its depth and a number of segments along its length. Uniaxial elements and engineering beam theory were used. An incremental approach was applied to calculate the stresses, strains and deflections in the time domain. The power creep law or the hyperbolic sine creep law was employed to compute the creep strain increment in each time interval. A computer program was written to carry out the computation. Comparison of numerical results with experimental results indicated good agreement when distinct properties for elements in tension and compression were used. A tran- sition zone was found in the stress distribution along the cross section. In the transition zone, the neutral axis moved up cu: down as time elapsed. and accordingly the stress and strain changed from tension to compresion or vice versa. Singularity was encountered near the neutral axis when the strain hardening theory was used. Such singularity. when it occurred, was avoided by use of the time hardening theory. In any case, the numerical results were not sensi- tive to the choice between the time hardening theory and the strain hardening theory. Two-Dimensional Finite Element Analysis: Finite element models involving multi-axial stress states were also developed for reinforced frozen soil. The formulation of 197 frozen soil elements followed the displacement method of finite element formulation of quadrilateral iSOparametric elements for homogeneous isotropic materials. To generate creep strains in the multi-axial stress state from the uni— axial creep relationship, the von Mise criterion and Plandtl-Ruess flow rule were used. The effective creep strain rate was computed from the effective stress by using uniaxial creep parameters. Since material properties of frozen soil in tension and compression are not the same, difficulty arises when both tension and compression are encountered. A weighing procedure, which evaluates the creep parameters in accordance with the three principal stresses, was proposed. Numerical results showed that these weighting functions worked well in comparing the numerical results with experimental data. The bond behavior between reinforcement and frozen soil was modeled by bond interface elements. The model was applied to reinforced frozen soil beams and to an analysis of the behavior of a system with an embedded reinforcing bar being “pulled out” from a frozen soil mass. A computer pro- gram was written to carry out the computation. Numerical results obtained from the computer model of the "pull out“ 'case compared well with experimental data. 5.2. Concluding Remarks The experimental method developed in Chapter 11 may be reproduced in any structural or soil laboratory. The equip- ment required can be easily made and the test procedures can 198 be carried out by one person. it may be standardized for testing the flexural behavior of frozen soil. The finding of a bilinear relationship in a log-log plot from reinforced frozen sand beam tests results is sig— nificant. The critical load, occurring at the break of the bilinear curve, is important in the design of frozen soil structures. Design engineers should always keep the applied load less than the critical load or the structure will behave as an unreinforced structure and the advantage of reinforcing will be lost. The need to use different creep parameters in tension and compression in the analysis of frozen soil beams in Chapter ill is a special characteristic different from beam creep analysis of other materials. It is important since the use of average creep parameters over tension and com- pression values leads to quite erroneous solutions. A major contribution in Chapter IV is the development of the weighing functions to circumvent the difference in material properties in tension and compression when using the effective stress and effective strain rate approach in a multi-axial stress case. With those weighting functions, a set of effective creep parameters can be evaluated from those in uniaxial tension and compression, and the frozen soil can be treated as a von Mise's material. Therefore, the use of more complicated yield criteria and the accompa- nying required material parameters (not yet available) may be avoided. 199 The bond interface element used in Chapter IV was modi- fied from the joint element developed by Goodman. et al. (1968) and the element proposed by Ghaboussi, et al. (1973) for linear material behavior. The interface element used here has incorporated the effect of creep. With the stress- relative displacement relationship pr0posed in Chapter IV. the bond behavior between the frozen soil and the structural member can be represented very well. This model should be applicable to many problems in cold region engineering such as pile foundations in frozen ground. A limitation of the numerical method described in Chapter III lies in the assumptions that the material is homogeneous and isotropic and plane sections remain plane. For small scale structures such as model frozen sand beams in Chapter ll. these assumptions would seem reasonable. For large scale structures in situ. materials may not always be homogeneous and plane sections may not remain plane after deformation. However. the method would still be adequate to estimate the creep deformations if the length of the member is much larger than its depth. Any nonhomogeneity of prop- erties can of course be taken care of by the finite element method. Extension of this study should include two aspects. In the experimental part, an attempt to measure the internal stress and strain should be included. Different types of frozen soil should be tested to study their flexural behavior. For the analysis, more sophisticated yield 200 criteria should be tried when necessary data for the mate- rial parameters become available. Three-dimensional finite elements may be developed to analyze bond behavior between the reinforcement and frozen soil. LIST OF REFERENCES LIST OF REFERENCES AKili, W. 1971. Stress-strain behavior of frozen fine- grained soils. Highway Res. Rec. 360:1-8. 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Proceedings of 2nd International Symposium of Ground Freezing, Trondheim, Preprints, pp. 109-119. Stoer, J. and Bulirsch R. 1980. Introduction to Numerical Analysis. Springer-Verlag Inc., New York Tapshell, H. J. and Johnson, A. E. 1935. An investigation of the nature of creep under stresses produced by pure flexure. J. Inst. of Metals 2:121-140. Thompson, E. G. and Sayles, F. H. 1979. In situ creep analysis of room in frozen soil. J. of Soil Mech. and Found., ASCE 9:899-915 Vialov, S. 5., ed. 1962. The strength and creep of frozen soils and calculations for ice-soil retaining structures. Trans. 76, U.S. Army, C.R.R.E.L., Hanover, NH. Weaver, J. S. and Morgenstern, N. R. 1981. Pile design in permafrost. Can. Geotech. J. 18:357-370. Zienkiewicz, 0. C. 1977. The finite element method. McGraw-Hill, London, England. Zienkiewicz, O. C. and Cormeau, I. C. 1974. Visco- plasticity, plasticity and creep in elastic solids - A unified numerical solution approach. Int'l J. for Num- erical Method in Engineering 8:821-845. APPENDIX A MODEL BEAM TESTS DATA APPENDIX A MODEL BEAM TEST DATA Abbreviations Position of Displacement Transducers: BM: Back Middle FM: Front Middle BL: Back Left BR: Back Right FL: Front Left FR: Front Right 207 Sample No. Type of Soil: Wedron Average Temperature: Weight of Sample: 1 (plain frozen sand beam) Weight of Soil: Weight of Water: 4. Sand Concentration: Dimension: Note: Sample No. 3.25"X2. 2 (plain frozen sand beam) Silica Sand —10°c .14 lbs lbs 24 lbs 64% 75xu0" leakage occured before test began. Type of Soil: wedron Silicg Sand Average Temperature: Weight of Sample: 26.14 lbs -10.1 C Weight of Soil: 21.9 lbs Weight of Water: 4.24 lbs Sand Concentration: 64% Dimension: Applied Load: 150 lbs 3.25"x2.75"x40" Temperature: -10.1 C Deflection: (inches) Elapsed Time(hr.) BM BL BR FM 0:00 .0032 ----- .0025 ----- 0:30 .0104 0074 .0063 0122 1:00 .0149 0081 .0081 0160 1:30 .0198 0104 .0105 0211 2:00 .0239 0120 .0116 0233 2:30 .0267 0148 .0140 0277 3:00 .0298 0163 .0169 0305 3:30 .0336 ----- .0200 0333 4:00 .0373 ----- .0226 0372 4:30 .0404 0249 .0250 .0388 5:00 .0436 .0269 .0270 .0422 5:30 .0479 .0333 .0288 .0461 6:00 .0519 .0360 .0295 .0500 6:30 .0553 .0390 .0328 .0527 6:40 .0604 .0462 .0350 .0600 Failure occured. Note: leakage was found after test ended. 209 Sample No. 3 (plain frozen sand beam) Type of Soil: Wedron Silicg Sand Average Temperature: -10.3 C Weight of Sample: 26.14 lbs Weight of Soil: 21.9 lbs Weight of Water: 4.24 lbs Sand Concentration: 64% Dimension: 3.25"x2.75"x40" Applied Load: 75 188 Temperature: ~10.l C Deflection: (inches) Elapsed TimeLhr.) BR BM 0:00 .0000 .0000 0:30 ----- .0032 1:00 ----- .0043 2:00 0012 .0063 3:00 0015 .0085 4:00 ----- .0095 5:00 0016 .0106 6:00 0018 .0112 7:00 0019 .0120 9:00 .0021 .0122 16:00 .0044 .0122 18:00 .0046 .0122 24:00 .0056 .0122 Applied Load: lOlolbs Temperature: 10.3 C Deflection: (inches) Elapsed Time(hr.) BL BR FM BM FL 0:00 .0056 .0056 .0122 .0122 0056 1:00 .0062 ----- .0133 .0132 ----- 2:00 .0069 .0065 .0147 .0141 0072 3:00 .0073 .0075 .0148 .0145 0079 4:00 ----- 0078 .0157 .0159 ----- 5:00 .0083 ----- .0171 .0160 0086 6:00 0087 0086 .0181 .0162 0089 7:00 .0095 ----- .0185 .0170 ----- 8:00 0096 0094 .0188 .0171 .0102 Note: positions of displacement transducers were rearranged between the end of the first load (75 lbs) and the begainning of the second load (101 lbs). 210 Applied Load: lOlOlbs (Sample no. 3 cont'd.) Temperature: 10.3 C Deflection: (inches) Elapsed Time(hr() BL BR FM BM FL 9:00 --------------- .0176 .0109 10:00 --------------- .0177 .0112 11:00 --------------- .0181 .0119 12:00 .0110 .0105 0190 .0185 ----- 13:00 .0115 .0120 0195 .0190 .0122 15:00 .0119 .0131 0206 .0193 0126 16:00 .0121 .0131 .0210 .0194 ----- 19:00 .0127 .0131 0213 .0203 0142 Applied Load: 127001 lbs Temperature: 10.3 C Deflection: (inches) Elapsed Time(hr.) BL BR FM BM FL 0:00 .0127 .0131 0213 .0203 0142 0:30 .0139 ----- 0236 .0224 ----- 1:00 --------------- .0236 0150 2:00 .0155 .0159 0267 .0248 0153 3:00 .0160 ----- 0276 .0256 0167 4:00 .0163 .0169 0289 .0265 0177 5:00 .0166 ----- 0294 .0275 0186 6:00 .0181 .0176 0317 .0284 ----- 7:00 ---------- 0319 .0293 0192 9:00 .0198 .0191 0326 .0302 0212 11:00 .0208 .0199 0345 .0312 .0220 12:00 --------------- .0317 0225 13:00 --------------- .0321 0230 14:00 .0223 ----- 0365 .0324 ----- 14:30 .0224 .0206 0369 .0326 0242 Applied Load: 153010 lbs Temperature: 10.3 C Deflection: (inches) Elapsed ' TimeLhr. ) BL BR FM BM FL 0:00 .0224 .0206 .0369 .0320 0242 0:30 .0238 ----- .0381 .0347 ----- 1:00 .0264 .0255 .0408 .0355 0259 2:00 .0291 .0281 .0437 .0374 0268 3:00 .0301 ----- .0456 .0387 ----- 4:00 .0312 ------ .0475 .0401 ----- 5:00 .0322 .0291 .0491 .0419 ----- 6:00 .0332 ----- .0504 .0431 ----- 6:30 .0336 .0300 .0506 .0437 .0331 211 Applied Load: 178.98 lbs (Sample no. 3 cont'd.) Temperature: -10.3 C Deflection: (inches) Elapsed * Time(hr.) BL BR FM BM FL 0:00 .0336 .0506 .0437 0331 0:30 .0350 .0532 .0465 0349 1:00 .0366 .0540 .0481 0363 2:00 .0395 .0570 .0509 0388 3:00 .0410 .0595 .0527 ----- 4:00 .0429 .0622 .0545 ----- 5:00 .0453' .0646 .0562 .0442 6:00 .0459 .0666 .0576 .0460 6:30 .0462 .0676 .0585 .0469 *Note: Recorder out of order. Applied Load: 205.81 lbs Temperature: -10.2 C Deflection: (inches) Elapsed Time(hr.) BL BR FM BM FL 0:00 .0462 .0676 .0585 0469 0:30 .0492 .0694 .0620 ----- 1:00 .0503 .0705 .0644 0496 2:00 .0530 .0721 .0673 0534 3:00 .0549 .0735 .0712 0561 4:00 .0571 .0759 .0736 0573 5:00 ----- .0763 .0758 .0587 6:00 ---------- .0771 .0602 7:00 .0644 .0784 .0786 0623 8:00 ---------- .0802 0639 8:30 .0663 .0804 .0806 0648 Applied Load: 231.81 lbs Temperature: -10.3 C Deflection: (inches) Elapsed Time (hr. ) BL BR FM BM FL 0:00 .0663 .0804 .0806 0648 0:30 .0707 .084 .0850 ----- 1:00 .0732 .086 .0872 0676 2:00 .0765 .0884 .0913 .0700 3:00 ---------- .0942 0728 4:00 ---------- .0979 0752 5:00 ---------- .1000 0772 6:00 ---------- .1014 0792 7:00 ---------- .1025 .0808 8:00 .0881 .1054 .1038 .0812 212 Applied Load: 257.11 lbs (Sample no. 3 cont'd.) Temperature: -10.3 C Deflection: (inches) Elapsed Time(hr.) BL FM BM FL 0:00 .0881 .1054 .1038 .0812 0:30 .0910 .1094 .1062 ----- 1:00 .0917 .1114 .1037 .0840 2:00 .0939 .1139 .1097 .0859 3:00 .0972 .1174 .1125 .0868 4:00 .1019 .1209 .1173 .0906 5:00 .1063 .1234 .1217 .0929 6:00 .1090 .1259 .1248 .0953 6:30 .1110 .1274 .1268 .0962 Applied Load: 282.85 lbs Temperature: -lO.2 C Deflection: (inches) Elapsed Time(hr.) BL BR FM BM FL 0:00 .1110 .1274 .1268 0962 0:30 .1139 .1309 .1328 ----- 1:00 ---------- .1366 1056 2:00 .1240 .1414 .1417 1107 3:00 .1277 .1454 .1454 1155 4:00 .1317 .1509 .1491 1206 5:00 .1365 .1559 .1520 1225 6:00 .1386 .1579 .1557 1254 6:10 .1394 .1587 .1559 1255 Applied Load: 308.35 lbs Temperature: ~10.2 C Deflection: (inches) Elapsed Time (hr. 1 BL BR FM BM FL 0:00 .1394 .1587 .1559 1255 0:30 .1441 .1642 .1647 ----- 1:00 .1456 .1662 .1700 .1358 2:00 .1630 .1872 .1824 .1490 3:00 .1746 .2012 .1956 .1631 4:00 .1852 .2097 .2079 .1762 5:00 .1940 .2197 .2185 .1824 6:00 ---------- .2304 .1914 7:00 ---------- .2406 .1970 8:00 ---------- .2503 .2064 9:00 .2352 .2657 .2640 2173 10:00 .2381 .2697 .2640 2173 11:00 .2440 .2777 .2732 2234 Applied Load: 334.80 lbs (Sample no. 3 cont'd.) Temperature: -10.2 C Deflection: (inches) Elapsed Time(hr.) BL BR FM BM FL 0:00 .2440 .2777 .2732 .2234 1:00 .2581 .2977 .2908 .2365 2:00 .2675 .3090 .3045 .2460 2:30 .2740 .3175 .3129 ----- 3:00 .2816 .3240 .3194 .2601 4:00 .2910 .3390 .3314 .2700 5:00 .3028 .3527 .3435 2813 6:00 .3175 .3727 .3602 2916 Applied Load: 360.33 lbs Temperature: -10.0 C Deflection: (inches) Elapsed Time(hr.) BL BR FM BM FL 0:00 .3175 .3727 .3602 .2916 1:00 -3351 -3977 ~3815 ~3057 2:00 .3457 .4115 .3983 .3156 3:00 .3539 .4252 .4122 .3245 4:00 .3645 .4377 .4242 .3335 5:00 ---------- .4345 .3391 6:00 .3763 .4602 .4438 .3461 6:30 .3810 .4702 .4484 .3496 Note: test stoped because of excess deformation. Sample No. 4 (plain frozen sand beam) Type of Soil: Wedron Sili8a Sand Average Temperature: -6.0 0 Weight of Sample: 26.14 lbs Weight of Soil: 21.9 lbs Weight of Water: 4.24 lbs Sand Concentration: 64% Dimension: 3.25"x2.75"x40" Applied Load: 70.03 lbs Temperature: -5.95 C Deflection: (inches) Elapsed Time(hr.) FL BL BR BM PM 0:00 .0000 .0000 .0000 .0000 .0000 0:30 .0026 .0024 .0018 .0027 .0030 1:00 .0048 .0035 .0029 .0046 .0052 2:00 .0070 .0058 .0050 .0072 .0072 214 Applied Load: 70.03 lbs (Sample no. 4 cont'd.) Temperature: -5.95 C Deflection: (inches) Elapsed Time(hr.)FL BL BR BM FM 3:00 .0089 .0081 .0061 .0094 .0097 4:00 .0109 .0104 .0075 .0118 .0119 5:00 .0122 .0117 .0084 .0138 .0138 6:00 .0132 .0127 .0091 .0159 .0158 7:00 .0156 .0153 .0109 .0175 .0172 8:00 .0168 .0167 .0117 .0190 .0194 9:00 .0188 .0190 .0135 .0202 .0216 10:00 .0202 .0202 .0147 .0217 .0244 11:00 .0222 .0224 .0167 .0229 .0275 Applied Load: 96.10 lbs Temperature: -6.1 C Deflection: (inches) Elapsed Time(hr.)FL BL BR BM PM 0:00 .0222 .0224 .0167 .0229 .0275 0:30 .0225 .0261 .0197 .0307 .0322 1:00 .0275 .0284 .0215 .0335 .0335 2:00 .0312 .0328 .0251 .0397 .0394 3:00 .0342 .0367 .0284 .0445 .0436 4.00 .0360 .0394 .0300 .0475 .0472 5:00 .0376 .0411 .0315 .0508 .0500 6:00 .0400 .0450 .0355 .0538 .0541 7:00 .0423 .0462 .0376 .0573 .0575 Applied Load: 122005 lbs Temperature: -6.1 C Deflection: (inches) Elapsed Time(hr.)FL BL BR BM FM 0:00 .0423 .0462 .0376 .0573 .0575 0:30 .0451 .0508 .0419 .0648 .0634 1:00 .0485 .0545 .0457 .0694 .0693 2:00 .0537 .0597 .0514 .0769 .0770 3:00 .0583 .0665 .0564 .0836 .0832 4:00 .0629 .0703 .0621 .0900 .0906 5:00 .0666 .0743 .0651 .0950 .0961 6:00 .0721 .0780 .0703 .1010 .1024 7:00 .0765 .0837 .0753 .1064 .1068 8:00 .0798 .0864 .0780 .1117 .1127 215 Applied Load: 148.14 lbs (Sample no. 4 cont'd.) Temperature: -6.1 C Deflection: (inches) Elapsed Time(hr.) FL BL BR BM FM 0:00 .0798 .0864 .0780 .0117 .1127 0:30 .0843 .0922 .0835 .1201 .1175 1:00 .0873 .0958 .0879 .1259 .1226 2:00 .0948 .1039 .0967 .1343 .1300 3:00 .1023 .1118 .1050 .1428 .1370 4:00 .1083 .1178 .1110 .1058 .1432 5:00 .1143 .1242 .1178 .1588 .1499 6:00 .1191 .1298 .1242 .1641 .1568 7:00 .1233 .1345 .1283 .1699 .1635 8:00 .1278 .1389 .1327 .1758 .1679 9:00 .1323 .1427 .1358 .1794 .1734 Applied Load: 173.89 lbs Temperature: -5.95 C Deflection: (inches) Elapsed Time(hr.) FL BL BR BM PM 0:00 .1323 .1427 .1358 .1794 .1734 0:30 .1401 .1502 .1468 .1901 .1815 1:00 .1452 .1547 .1512 .1971 .1881 2:00 .1539 .1627 .1605 .2087 .1981 3:00 .1616 .1700 .1694 .2194 .206 4:00 .1704 .1777 .1798 .2296 .219 5:00 .1792 .1852 .1880 .2407 .2220 6:00 .1869 .1930 .1963 .2505 .2286 7:00 .1969 .2027 .2051 .2616 .2360 Applied Load: 199.87 lbs Temperature: -5.95 C Deflection: (inches) Elapsed Time(hr.) FL BL BR BM FM 0:00 .1969 .2027 .2051 .2616 .2360 0:30 .2080 .2132 .2167 .2729 .2480 1:00 .2139 .2187 .2216 .2813 .2555 2:00 .2257 .2297 .2343 .2982 .2685 3:00 .2369 .2402 .2452 .3104 .2805 4:00 .2469 .2477 .2535 .3235 .2905 5:00 .2557 .2557 .2596 .3366 .3005 6:00 .2686 .2639 .2689 .3460 .3095 7:00 .2780 .2729 .2749 .3553 .3180 8:00 .2845 .2795 .2854 .3675 .3280 216 Applied Load: 225.86 lbs (Sample no. 4 cont'd.) Temperature: -6.05 C Deflection: (inches) Elapsed Time(hr.) FL BL BR BM FM 0:00 .2845 .2795 .2854 .3675 .3280 0:30 .2935 .2875 .2921 .3806 .3440 1:00 .2995 .2945 .2959 .3891 .3515 2:00 .3130 .3035 .3041 .4012 .3665 2:30 .3194 .3085 .3104 .4097 .3745 3:00 .3250 .3165 .3154 .4162 .3810 4:00 .3359 .3215 .3241 .4275 .3940 5:00 .3424 .3325 .3291 .4378 .4040 6:00 .3490 .3425 .3379 .4463 .4120 7:00 .3565 .3495 .3479 .4566 .4200 8:00 .3635 .3535 .3516 .4631 .4275 9:00 .3690 .3585 .3567 .4734 .4345 Note: Test stoped due to excess deformation. Sample No. 5 (reinforced frozen sand beam) Type of Soil: Wedron Silica Sand Average Temperature: -10.1 0 Weight of Sample: 27.86 lbs Weight of Soil: 22.795 lbs Weight of Water: 4.575 lbs Sand Concentration: 64% Dimension of Beam: 3.25"x2.75”x40" Reinforcement: a smooth steel bar Diameter of Steel Bar: 3/16" Length of Steel Bar: 40" Weight of Steel Bar: 0.31 lb Position of Steel Bar: 0.5” from top of the beam to the top of the steel bar. Applied Load: 96.16 lbs Temperature: -10.5 C Deflection: (inches) Elapsed Time (hr . ) FL QR BL BM FM 0:00 .0000 .0000 0000 .0000 0000 0:30 .0009 .0006 0009 .0009 ----- 1:00 ----- .0007 ----- .0012 ----- 2:00 0011 .0011 ----- .0019 ----- 5:00 0014 .0018 0013 .0035 ----- 8:00 0018 .0027 0016 .0042 ----- 11:00 0019 .0033 .0020 .0051 ----- 15:00 0021 .0039 0024 .0058 ----- 18:00 0023 .0041 0026 .0061 0061 217 Applied Load: 122'18 lbs (Sample no. 5 cont'd.) Temperature: -10.15 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM FM 0:00 .0023 .0041 .0026 .0061 0061 0.30 .0029 .0048 .0033 .0069 ----- 1:00 .0032 .0051 .0037 .0070 ----- 2:00 .0035 .0052 .0040 .0072 ----- 4:00 .0037 .0055 .0043 .0078 0075 6:00 .0042 .0060 .0049 .0086 ----- 8:00 .0046 .0063 .0054 .0089 0089 10:00 .0049 ----- .0056 .0094 ..... 12.00 .0052 .0073 .0059 .0099 ----- 14:00 .0056 .0075 .0069 .0102 ----- 15:00 .0064 .0075 .0076 .0105 0104 Applied Load: 147.96 lbs Temperature: -10.05 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM PM 0:00 .0064 .0075 .0076 .0105 0104 0:30 .0068 .0079 .0079 .0111 ----- 1:00 -0076 ----- .0083 .0115 ----- 2:00 .0081 .0083 .0087 .0118 0121 3:00 .0086 .0086 .0090 .0123 ----- 4:00 .0092 .0094 .0096 .0127 0127 7:00 .0104 .0111 .0106 .0135 ----- 8:00 .0106 .0114 .0108 .0138 0138 9:00 .0110 .0119 .0111 .0141 ----- 10:00 .0114 .0123 .0116 .0143 0141 11:00 .0115 .0124 .0117 .0146 ----- 12:00 .0116 .0124 .0118 .0149 ----- 12:30 .0118 .0125 .0119 .0150 0148 Applied Load: 173.89 lbs Temperature: -10.15 C Deflection: (inches) Elapsed Time(hr.) FL BR w13L BM FM 0:00 .0118 .0125 .0119 .0150 0148 0:30 .0124 .0132 .0125 .0156 ----- 1:00 .0127 .0134 .0128 .0160 ----- 2:00 .0134 .0139 .0135 .0164 0162 3:00 .0138 .0142 .0136 .0168 ----- 4:00 .0143 .0146 .0140 .0173 0170 5:00 .0148 .0150 .0147 .0177 ----- 6:00 .0150 .0154 .0149 .0179 0174 7:00 .0155 .0158 .0152 .0181 ----- 7:30 .0157 .0159 .0155 .0185 0179 218 Applied Load: 225.88 lbs (Sample no. 5 cont'd.) Temperature: -10.1 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM PM 0:00 .0157 .0159 .0155 .0185 0179 0:30 .0168 .0178 .0176 .0205 ----- 1:00 .0175 .0186 .0184 .0213 0202 2:00 .0187 .0201 .0198 .0225 0207 3:00 .0200 .0216 .0211 .0235 0225 4:00 .0204 .0222 .0216 .0240 0230 5:00 .0213 .0230 .0226 .0247 0236 6:00 .0223 .0239 .0238 .0252 0242 7:00 .0227 .0241 .0241 .0258 0245 Applied Load: 277.81 lbs Temperature: -10.1 C Deflection: (inches) Elapsed * Time(hr.) FL BR BL BM FM 0:00 .0227 .0241 .0241 .0258 0:30 .0249 .0254 .0268 .0304 1:00 .0258 .0285 .0282 .0322 2:00 .0274 .0308 .0298 .0351 3:00 .0290 .0327 .0314 .0374 4:00 .0306 .0337 .0334 .0387 5:00 .0316 .0353 .0342 .0402 6:00 .0330 .0378 .0355 .0415 *Note: displacement transducer out of order. Applied Load: 329.68 lbs Temperature: -10.25 C Deflection: (inches) Elapsed Time Lhr.) FL BR BL BM FM 0:00 .0330 .0378 .0355 .0415 0:30 .0355 .0399 .0390 .0456 1:00 .0368 .0417 .0408 .0471 2:00 .0398 .0445 .0436 .0500 3:00 .0416 .0469 .0458 .0523 4:00 .0433 .0486 .0479 .0544 5:00 .0454 .0506 .0499 .0561 6:00 .0470 .0523 .0522 .0580 219 Applied Load: 381.37 lbs (Sample no. 5 cont'd.) Temperature: ~10.2 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM FM 0:00 .0470 .0523 .0522 .0580 0:30 .0499 .0554 .0552 .0628 1:00 .0511 .0575 .0575 .0652 2:00 .0536 .0615 .0608 .0696 3:00 .0563 .0643 .0639 .0723 4:00 .0583 .0660 .0662 .0751 5:00 .0601 .0673 .0680 .0781 6:00 .0622 .0697 .0704 .0811 Applied Load: 433.10 lbs Temperature: -l0.0 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM PM 0:00 .0662 .0697 .0704 .0811 0:30 .0669 .0747 .0748 .0861 1:00 .0696 .0795 .0778 .0898 2:00 .0747 .0847 .0838 .0950 3:00 .0791 .0897 .0893 .1005 4:00 .0843 .0947 .0959 .1058 5:00 .0894 .0997 .1019 .1118 5:50 .0932 .1020 .1064 .1164 Applied Load: 459.10 lbs Temperature: -10.0 C Deflection: (inches) Elapsed Time Lhn) FL BR BL 13M FM 0:00 .0932 .1020 .1064 .1164 Note: failure due to bond slip occured at the right support immediately after the load increment (26 lbs) was added. Sample No. 6 (reinforced frozen sand beam) Type of Soil: Wedron Silicg Sand Average Temperature: -10.1 C Weight of Sample: 27.60 lbs Weight of Soil: 22.445 lbs Weight of Water: 4.825 220 Sand Concentration: 63.72% (Sample no. 6 cont'd.) Dimension of Beam: 3.25"x2.75"x40" Reinforcement: a steel bar with lugs at both ends Diameter of Steel Bar: 3/16" Length of Steel Bar: 40" Weight of Steel Bar: 0.33 lb. Lug Diameter: 3/16" Lug Length: 1/2" Lug position: l/2" from both ends of the steel bar Position of Steel Bar: 1/2" from the top of steel bar to the top of the beam. Applied Load: 148.85 lbs Temperature: -l0.0 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM FM 0:00 .0000 .0000 0000 .0000 .0000 0:30 .0005 .0009 ----- .0018 .0025 1:00 .0014 .0016 0015 .0030 .0050 2:00 .0028 .0029 ----- .0048 .0070 3:00 .0038 .0043 0039 .0058 .0100 4:00 .0051 .0060 ----- .0068 .0105 5:00 .0054 .0067 ----- .0080 .0110 6:00 .0059 .0078 0045 .0086 .0115 7:00 .0074 .0092 0053 .0108 .0125 8:00 .0078 .0103 0064 .0118 .0135 9:00 .0083 .0110 0069 .0128 .0145 10:00 .0090 .0123 0078 .0133 .0150 11:00 .0097 .0133 0091 .0142 .0155 11:50 .0100 .0137 0096 .0150 .0165 Applied Load: 199.73 lbs Temperature: -10.05 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM FM 0:00 .0100 .0137 .0096 .0150 .0165 0:30 .0124 .0168 .0125 .0196 .0209 1:00 .0139 .0182 .0139 .0215 .0221 2:00 .0152 .0200 .0154 .0233 .0259 3:00 .0159 .0212 .0167 .0254 .0271 4:00 .0171 .0224 .0182 .0273 .0290 5:00 .0180 .02 7 .0192 .0285 .0321 6:00 .0186 .02 8 .0199 .0296 .0334 7:00 .0194 .0258 .0209 .0315 .0340 8:00 .0200 .0266 .0215 .0337 .0358 9:00 .0208 .0270 .0225 .0358 .0365 10:00 .0221 .0279 .0239 .0379 .0371 10:30 .0227 .0281 .0246 .0392 .0378 221 Applied Load: 251.33 lbs (Sample no. 6 cont'd.) Temperature: -lO.4 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM FM 0:00 .0227 .0281 .0246 .0392 .0378 0:30 .0268 .0314 .0282 .0430 .0409 1:00 .0282 .0328 .0299 .0445 .0421 2:00 .0305 .0353 .0324 .0467 .0446 3:00 .0320 .0374 .0346 .0490 .0459 4:00 .0341 .0394 .0367 .0511 .0484 5:00 .0352 .0411 .0383 .0520 .050 6:00 .0368 .0428 .0,2 .0540 .052 6:30 .0375 .0434 .0415 .0545 .0534 Applied Load: 303.86 lbs Temperature: -10.1 C Deflection: (inches) Elapsed Time(hrL) FL BR BL BM FM 0:00 .0375 .0434 .0415 .0545 .0534 0:30 .0425 .0465 .0472 .0584 .0590 1:00 .0459 .0499 .0510 .0621 .0617 2:00 .0497 .0532 .0541 .0661 .0659 3:00 .0519 .0564 .0572 .0709 .0700 4:00 .0550 .0595 .0605 .0739 .0717 5:00 .0571 .0622 .0630 .0774 .0759 6:00 .0597 .0654 .0645 .0802 .0792 7:00 .0613 .0679 .0658 .0834 .0817 Applied Load: 355.18 lbs Temperature: -10.1 C Deflection: (inches) Elapsed Time(hr.) FL BR BL 3_§M PM 0:00 .0613 .0679 .0658 .0834 .0817 0:30 .0659 .0732 .0706 .0894 .0897 1:00 .0676 .0752 .0735 .0919 .0917 2:00 .0722 .0794 .0775 .0972 .0947 3:00 .0751 .0834 .0809 .1022 .0997 4:00 .0779 .0868 .0852 .1059 .1027 5:00 .0816 .0900 .0895 .1099 .1067 6:00 .0841 .0925 .0929 .1137 .1117 7:00 .0872 .0952 .0958 .1172 .1137 8:00 .0901 .0973 .0995 .1211 .1187 9:00 .0931 .0992 .1020 .1249 .1217 10:00 .0963 .1005 .1043 .1284 .1247 222 Applied Load: 407.13 lbs (cont'd.) Temperature: -10.15 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM FM 0:00 .0963 .1005 .1043 .1284 .1247 0:30 .1016 .1037 .1083 .1334 .1309 1:00 .1041 .1058 .1101 .1359 .1347 2:00 .1088 .1094 .1157 .1424 .1397 3:00 .1144 .1152 .1215 .1484 .1460 4:00 .1213 .1205 .1277 .1549 .1497 5:00 .1276 .1268 .1357 .1604 .1572 6:00 .1338 .1320 .1422 .1674 .1672 7:00 .1401 .1372 .1500 .1734 .1747 Applied Load: 459.06 lbs Temperature: -lO.l5 C Deflection: (inches) Elapsed Time(hr.) FL BR BL BM PM 0:00 .1401 .1372 .1500 .1734 .1747 0:30 .1494 .1450 .1602 .1809 .1872 1:00 .1556 .1498 .1660 .1889 .1935 2:00 .1663 .1598 .1775 .2039 .2047 3:00 .1775 .1698 .1872 .215 .2159 4:00 .1900 .1792 .1991 .228 .2259 5:00 .1993 .1897 .2100 .2399 .2385 6:00 .2125 .1976 .2158 .2499 .2497 7:00 .2225 .2159 .2232 .2604 .2610 Applied Load: 511.03 dbs Temperature: -10.15 C Deflection:4(inches) Elapsed Time (hrg FL BR BL BM FM 0:00 .2225 .2159 .2232 .2604 .2610 0:30 .2361 .2268 .2375 .2754 .2777 1:00 .2429 .2330 .2460 .2874 .2911 2:00 .2583 .2492 .2646 .3129 .3155 3:00 .2742 .2589 .2746 .32ZZ .3362 4:00 .2837 .2697 .2860 .34 .3495 5:00 .3028 .2863 .3070 .3579 .3662 6:00 .3173 .3009 .3199 .3774 .3829 7:00 -3317 .3123 -3351 -3939 -3996 8:00 .3363 .3269 .3499 .4074 .4146 9:00 .3610 .3387 .3643 .4239 .4297 10:00 .3756 .3520 .3803 .4404 .4447 10:14 .3795 .3561 .3824 .4429 .4464 Note: failure occured due to bond slip. APPENDIX B COMPUTER PROGRAM FOR ONE-DIMENSIONAL FINITE ELEMENT ANALYSIS 2233 RRDCRAM EzNSOILIINPUT.OUTpUT,TARE 5=INPUT.TAPE 6=OUTRUTI 100 C 110 C KEY--- 120 : TITLE- NAME 0: RROCRAM :30 C BL ----- BEAM LENGTH. 140 c 8h ----- BEAM HEIGHT. :50 c 8W ----- BEAM wIDTM. 160 c NI ----- NUMBER OF NODES IN x DIRECTION 170 c NJ ----- NUMBER OF NODES IN v DIRECTION. 180 c xII)---x COORDINATES 0F NOOE I. 190 c v101—--v COORDINATES OF NODE J. 200 c NL ----- TOTAL NUMBER OF LOADING. 210 c LN ----- LOADING NUMBER. 220 C LTN----LOADINC TYPE NUMBER. 230 c LMACA--LOADINC MACNITUDE. 240 c LDOSI--LOADINC ROSITION. 250 c M(I)---8ENDING MOMENT or SECTION I. 260 C R(:1---AxIAL RRESSURE OF SECTION I. 270 c SIGMAC-RROOF STRESS.SICMA C. 280 c N ------ CREER RowER RARAMETER N. 290 C B ------ CREER powER RARAMETER B. 300 C Nc ----- CREEP RowER RARAMETER N FOR COMPRESSION. 3‘0 C NT ----- CREEP RowER RARAMETER N FOR TENSION 320 c C ----CREEP POWER PARAMETER B FOR COMRRESSION. 330 C BT ----CREER POWER RARAMETER B FOR TENSION. 340 C E(d)---INITIAL TANCENT MODULus 0F LAYER U. 350 c ECDOT--STRAIN RATE OF LAYER J.EPSILON C DOT. 360 c EC ----- CREEP STRAIN.EPSILON c. 370 C ET ----- TATAL STRAIN. ERSILON T. 380 c TIMEINL-INITIAL CREEP TIME. 390 C TIMEFNL-FINAL CREEP TIME. 400 C DELTAT-TIME INCREMENT.DELTA T. A10 C TRRINT-TIME To RRINT. 420 c ETEINAL-EINAL TENSION STRAIN. 430 C ECEINAL-EINAL COMPRESSION STRAIN. 440 c ETAIUi-DISTANCE FROM NEUTRAL AxIS. 450 c HIUI---THICKNESS OF LAYER U. 460 c Y21I)--CURVATURE OF SECTION I. 470 c K ------ TIME INCREMENT NUMBER. 480 c NRRINT-TOTAL NUMBER OF NODES wHICH STRESSES AND STRAIN TO BE PRINT . 490 c NODEPT- NODES wMICH STRESSES AND STRAIN To BE RRINT. 500 C 510 INTEGER LTN(51).NODEPT(51) 520 REAL X(51).Y(51).M (511.9(51) 530 REAL Y2(51).ET151.51). EC(51.511.SIGMA1(51,51) 540 REAL LMAGA(51).LPOSI(51).E(51).NC.NT 550 REAL DEFLEC(511.DEF(51) .LTIME151) 560 c 570 COMMON /8LOCK2/ DELTAT.E.SIGMACC(101.SIGMACT(10).NC(10). 580 + NT(10).8C(10).BT(10).ECDOTC(10).ECDOTT(10) 590 COMMON /BLOCK3/ LMACA.LROSI.LTIME soc COMMON /BLOCK4/ TIME 61o COMMON /BLOCKS/ v.TIMEINL.TIMEFNL.ECEINAL.ETEINAL.TRRINT 620 COMMON /8LOCK6/ x 630 COMMON /BLOCK8/ NPRINT,NODEPT 54o COMMON /BLOCK9/ EC 650 COMMON /BLDCK:O/ ECOMR(10).ETENS(10) 660 COMMON /BLOCK1/ DEFLEC.DEF 670 c 680 c ----------- INPUT DATA --------- 690 c 700 CALL OATAINP(BL.8H.BW.NI.Nd.NL.LTN) 710 C 720 TIMEINL-TIMEINL 730 TIMEENLsTIMEENL 740 DELTATsOELTAT 750 TRRINT-TRRINT 760 TIMERCCcTIMERcc 770 TIMERCT-TIMERCT 780 D0 3013 I-1.NL 790 LTIME(I)-LTIME(I) 800 3013 CONTINUE 810 LTIME(NL+1)- 0. B20 2224 x: 1 KP= o LN- 1 JUMP: o IPTsINTITRRINT/DELTAT) DT=DELTAT TIME . TIMEINL+DELTAT DO 3010 I‘1.NI DO 3009 d‘1.Nd SIGMA1(I.UD= O. EC(I.dl= 0. ETlI.d)= 0. 3009 CONTINUE Y211)‘ 0. M(I)‘ O. P(I)= O. DEFLEC(I)=O. 05‘111'0. 3010 CONTINUE C C C (wheat) ()Uan 0 C C C C 3004 CONTINUE IF((LTIME(LN).EO.(TIME-DELTAT)).ANO.(dUMP.E0.0)) DO 4001 I'1.NI OEF(I)'DEFLEC(I) 4001 CONTINUE THEN L2-LN 30:2 IEILTIMEIL2+1).E0.(TIME-DELTAT,1 THEN L2-L2+1 IF(L2.LT.NL) 50 TO 3012 END IF --------- COMRUTE MOMENT ANO AxIAL FORCE ----- CALL MOMENTINI.L2.LTN.BH.Bw.BL.LN.M,p1 -------- COMPUTE ELASTIC STRESS AND STRAIN ----- CALL STRESSINI.NJ.O.Bw. o.SICMA:.ET.v2.v.BL.M,p) LN-L2+1 TIME-TIME-DELTAT K-K-1 UUMR- : GO TO 3016 END IF ---------- COMPUTE CREER STRESSES AND STRAIN CALL STRESS(NI.NJ.K.Bw.KP.SICMA1.ET.Y2.V.BL.M.R) JUMR- O -------- COMPUTE DEFLECTION------- 3016 CALL OEFLECT(NI.BL.Y2) PRINT 3017. TIME PRINT 2002 --------- CHECK TIME To STOP------ DO 3002 I ' 1.NI DO 3005 d'1.NJ.Nd IF(ET(I.J).GE.0.)THEN IF(ABS(ET(I.d)).GE.ABS(ECFINAL)) GO TO 3003 830 840 850 860 870 880 890 910 920 930 940 950 960 970 980 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1450 1470 1430 1490 1500 1510 1520 1530 1540 1550 225 ELSE 1560 IF(ABS(ET(I.d)).GE.ABS(ETFINAL)) GO TO 3003 1570 END IF 1580 3005 CONTINUE 1590 3002 CONTINUE 1600 C 1610 IF(TIME.LT TIMEFNL)THEN 1620 TIME - TIME + OELTAT 1630 IF(((LTIME(LN)-TIME).GT.O.).ANO.((LTIME1LN)-TIME).LT.OT)) THEN 1640 OELTAT=LTIME(LN)-TIME 1550 ELSE 1660 OELTAT=DT 1570 END IF 1530 IF(K EO.KP) KP=K+IPT 1590 K=K+1 1700 GO TO 3004 1710 END IF 1720 2002 FORMAT(/1X.120(“-” )/) 1730 3017 FORMAT(/1x.'--- TIME s '.F1S.7.' HOURS---'/) 1740 C 1750 3003 END 1760 c 1770 SUBROUTINE OATLINP(BL.BH.BU.NI.Nd.NL.LTN) 1780 c 1790 C KEY-- SAME AS MAIN PROGRAM. 1800 C 1810 CHARACTER TITLE-80 1820 INTEGER NI.Nd.NL.LTN(51).NOOEPT(51I 1830 REAL 5L.BH.8w.X(51).Y(51).LMAGA(51).LPOSI(51).LTIME(51) 1840 REAL NC.NT.E(S1) 1850 C 1860 COMMON /8LOCK2/ DELTAT.E.SIGMACC(10).SIGMACT(10).NC(10). 1870 + NT(10).8C(10).BT(10).ECOOTC(10).ECOOTT(10) 1880 COMMON /BLOCK3/ LMAGA.LROSI.LTIME 1890 COMMON /BLOCK5/ Y.TIMEINL.TIMEFNL.ECFINAL.ETFINAL.TPRINT 1900 COMMON /BLOCK6/ x 1910 COMMON /BLOCK8/ NPRINT.NOOEPT 1920 COMMON /BLOCK10/ ECOMR(10).ETENS(10) 1930 COMMON /8;0CK7; CLAN.MTL(51) 1940 C 1950 c ---------- INRUT DIMENSIONS AND COORDINATED ----- 1960 C 1970 READ(S.1001) TITLE 1980 REA015.1002) 8L.8H.8w 1990 READ1S.1003) ICLAw 2000 READ1S.1003) NI 2010 REA015.1002) (XII).I-1.NI) 2020 READ15.1003) N0 2030 REAO(S.1002) (Ytu).u-1.Na) 2040 READ1E.1003) NL 2050 DO 3103 I I 1.NL 2060 REA015.1006) LTN(I).LMAGA(I).LPOSI(I).LTIME(I) 2070 3103 CONTINUE 2080 C 2090 C ---------- INPUT MATERIAL PROPERTIES ----- 2100 C 2110 READIS.1003) MAXMAT 2120 DO 3101 It1.MAXMAT 2130 REA015.1007) ECOMRII).ETENS(I) 2140 REAO(5.1007) SIGMACC(I).NC(I).BC(I).ECOOTC(I) 2150 REAO(5.1007) SIGMACT(I).NT(II.8T(I).ECOOTT(I) 2160 3101 CONTINUE 2170 REAO(5.1010) (MTL(d).d-1.Nd-1) 2180 REAO(5.1007) ECFINAL.ETFINAL 2190 REAO(S.1002) TIMEINL.TIMEFNL.DELTAT 2200 C 226 REAO(S.1002) TPRINT REAO(S.1003) NPRINT READ(5.1010) (NOOEPT(I),I=1,NPRINT) --------- PRINT DUT INTPUT DATA ------- PRINT 2003.TITLE -------- PRINT OUT BEAM DIMENSION ---—- PRINT 2004 PRINT 2005. BL PRINT 2006. EH PRINT 200?. BN PRINT 2037. ICLAw PRINT 2002 --------- PRINT OUT COORDINATE----- PRINT 2024 PRINT 2025.NI PRINT 2026 DD 3205 I-1.NI PRINT 2027. I.x(I) 3205 CONTINUE PRINT 2028.Nd PRINT 2026 DO 3206 0-1.NU PRINT 2027.0.Y(d) 3206 CONTINUE PRINT 2002 Nu-Nd-1 C ------------ PRINT OUT LOADING CONDITION -------- PRINT 2005 PRINT 2009. NL PRINT 2010 DO 3201 I . 1.NL IF(LTN(I).EO.1)PRINT 2011. I.LMAGAII).LPOSI(I).LTIME(I) IFT;TN(I).E0.21PRINT 2012. I.LMAGAII).LPOSI(I).LTIME(I) IFILTN111.E0.31PRINT 2013.1.LMAGAII).LTIME(I) IF1LTN(I).EO.4IPRINT 2014. I.LMAGA(I).LPOSI(I).LTIME(I) IF(LTN(I).EQ.51PRINT 2015. I.LMAGAII).LTIME(Z) 3201 CONTINUE c ---------- PRINT OUT MATERIAL PROPERTIES ----- , PRINT 2002 PRINT 2016 DO 3102 I-1.MAXMAT PRINT 2036. I PRINT 2017. SIGMACC(I) PRINT 2018. SIGMACT1Z) PRINT 2019. NC(I) PRINT 2020. NT(I) PRINT 2032. BC(I) PRINT 2033. BTII) PRINT 2031. ECOOTCII).ECDOTT(I) PRINT 2029. ECOMP(I) PRINT 2030. ETENS(I) 3102 CONTINUE PRINT 2034. ECFINAL PRINT 2035. ETFINAL PRINT 2021. TIMEINL PRINT 2022. TIMEFNL PRINT 2023. DELTAT PRINT 2002 C c ---------- INTPUT FORMAT ---------- C 1001 FORMAT1ABO) 1002 FORMAT18E10.5) 2210 2220 2230 2240 2250 2260 2270 2280 2290 2300 2310 2320 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 2580 2590 2600 2610 2620 2630 2640 2650 2660 2670 2680 2690 2700 2710 2720 2730 2740 2750 2910 2930 227 1003 FORMAT(I2) 1006 FORMAT(I1.9X.3F10.5) 1007 FORMAT(4E20.7) 1010 FORMAT(1615) C ---------- OUTPUT FORMAT ---------- C 2002 FORMAT(/1X.120(“-' )/) 2003 FORMAT(/1x,120("-”)/1x.A80 /1x,120("-“)) 2004 FORMAT(/1X.'BEAM DIMENSION"/) 2005 FORMAT(1x."BEAM LENGTH = ".F10.5.2x."INCHES") 2006 FORMAT (1x.'BEAM DEPTH . ”.F10.5.2X.”INCHES”) 2007 FDRMAT(1X."BEAM wIOTH . ',F10.5.2X.“INCHES“) 2008 FORMAT(/1X.“LOADING"/) 2009 FORMAT(1X.‘TOTAL NUMBER OF LOADING I “.I2/) 2010 FORMAT(1X.'LOADING NUMBER ', I'TYPE OF LOADING ‘, 'MAGNITUDE ", “DIMENSIONS ”. “LOCATION ', “LOADING TIME“/) 2011 FORMAT(10X.I2.9X.”POINT LOAO“.20X.E14.7.6X .“POUNDS“.14X.'X . ' +++++ + F10.5.2X.'INCHES".5X.F10.5) 2012 FORMAT(10X.I2.9X.”POINT MOMENT'.18X.E14.7.6X .‘POUND-INCHES". + 8X.”X 3 '.F10.5.2X.“INCHES“.SX.F10.5) 2013 FORMAT(10X.I2.9X.“UNIFORM DISTRIBUTED LOAD'.6X.E14.7.6X . + “POUNDS PER INCHES".5X.F10.5) 2014 FDRMAT(10X.I2.9X.'LINEAR DISTRIBUTED LOAD'.6X.E14.6.6X . + “LB/IN“.3X.“AXIAL LOAD AT X8 '.F10.5.2X. + 'INCHES'.1X.F10.5) 201s FORMAT(10X.I2.9X.'AXIAL LOAD'.1OX.E14.6.6X .‘POUNDS'.5X.F10.5) 2016 FORMAT(/1X,'MATERIAL PROPERTIES'/) 2017 FORMAT(1x.~PRODF STRESS FOR COMPRESSION - '.E20.7.2X.‘PSI') 2018 FORMAT(1x.-PRDOF STRESS FOR TENSION - '.E20.7.2X.'PSI') 2019 FORMAT(1X.“CREEP PowER PARAMETER N FOR COMPRESSION - '.F10.5) 2020 FDRMAT(1X.'CREEP PowER PARAMETER N FOR TENSION - '.F10.5) 2021 FORMAT(1X.'DESIGNATED INITIAL TIME . “.F10.5.2X.'HOURS“) 2022 FORMAT(1X.'OESIGNATED FINAL TIME . '.F10.5.2X.'HOURS') 2023 FORMAT(1X.“DESIGNATED TIME INCREMENT . '.F10.5.2X.‘HOURS') 2024 FORMAT(1X.'COORDINATE'/) 2025 FORMAT(1X.'TOTAL NUMBER OF NODES IN x DIRECTION - ".12/) 2026 FDRMAT(1x.-NODE NUMBER”.9X.'COORDINATE'/) 2027 FORMAT(1OX.I2.9X.F10.5) 2028 FORMAT1/1X.“TDTAL NUMSER OF NODES IN v DIRECTION . “.12/1 2029 FORMAT(/1X.”ELASTIC MODULUS FOR COMPRESSION - ".E1s.7.~ PSI“) 2030 FORMAT1/1x."ELASTIC MOOULUS FOR TENSION - “.E15.7.“ PSI'/| 2031 FORMAT1/1X.“°ROOF STRAIN RATE FOR COMPRESSION ' “.520.7.” (1/HR)” + /1x.-PROOF STRAIN RATE FOR TENSION - ".E20.7.' (1/HR)"/) 2032 FORMAT11x."CREEP PowER PARAMETER 8 FOR COMPRESSION . ".F10.5) 2033 FORMAT(1X.“CREEP PDwER PARAMETER 8 FOR TENSION - '.F10.S) 2034 FORMAT(1X.'DESIGNATED FINAL STRAIN FOR COMPRESSION - '.F1C.5) 2035 FORMAT11x.'DESIGNATED FINAL STRAIN FOR TENSION . '.F10.S) 2036 FORMAT(//1X.'MATERIAL PROPERTv NUMBER - “.IS//) 2037 FORMAT(/1X.“CREEP LAw CODE - “.15. + /6X."Eo.0. PONER LAN. TIME HARDENING'. + /6X.'EO.1. PONER LAN. STRAIN HARDENING ' + /6X.'E0.2. HYPOBOLIC SIN LAN“) END 2940 2950 2960 2970 2980 2990 3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100 3110 3120 3130 3140 3150 3160 3170 3180 3190 3200 3210 3220 3230 3240 3250 3260 3270 3280 3290 3300 3310 3320 3330 3340 3350 3360 3370 3380 3390 3400 3410 3420 3430 3440 3450 3460 3470 3480 3490 3500 3510 3520 228 C SU8RDUTINE MOMENTINI.L2.LTN.8H.8w.L.LN.M.P) C C KET~- C L ------ BEAM LENGTH. C MM(I)--8ENOING MOMENT OF SECTION I FROM ONE LOADING ONLY. C PP1I)--AAIAL PRESSURE OF SECTION I EROM ONE LOADING ONLY. C OTHERS-SAME AS MAIN PROGRAM. INTEGER LTN1511 REAL L.LMAGA1611.LPDSI(51).LTIME1511 REAL M151),P(611.x(611.MM151).PP151) C COMMON /8LOCK3/ LMAGA.LPOSI.LTIME COMMON IBLOCK6/ x C C --------- COMPUTE BENDING MOMENT AND AXIAL PRESSURE--- C c 00 3201 L1=LN.L2 C 00 3212 I = 1.NI MM(I) - 0. PP(I) . 0. 3212 CONTINUE IF(LTN(L1).EO.1)THEN DO 3202 I = 1.NI IF(X(I).LE.LPDSI(L1))THEN MM(I) - LMAGA(L1)-(L-LPOSI(L1))-x(I)/L ELSE MM(I) - LMAGA(L1)-(L-LPOSI(L11)-x(I)/L-LMAGA(L1)-(X(I)- + LPOSI(L1)) END IF 3202 CONTINUE END IF c IF(LTN(L1).EO.2)THEN DO 3203 I - 1.NI IF((X(I).GT.LPOSI(L1)).OR.(LPDSI(L1).E0.0.))THEN MM(I) - LMAGA(L1)-X(I)/L-LMAGA(L1) ELSE MM(I) - LMAGA(L1)'X(I)/L END IF 3203 CONTINUE END IF C IF(LTN(L1).E0.3)THEN DO 3204 I-1.NI MM(I)-LMAGAIL1)-(L-x(1)-XII)--2)/2. 3204 CONTINUE END IF C IF(LTN(L1).EO.4)THEN DO 3205 I - 1.NI IF(x(I).LT.LPOSI(L1))THEN MM(I)-LMAGA(L1)-X(I)-(2.-L-LPOSI(L1)~X(I)--2/ + LPOSI(L1)) /6. ELSE IF(x(I).Eo.L)THEN MM(I)- 0. ELSE MM(I)=LMAGA(L1)-(L-x(I))-(L+LPOSI(L1)-(L-x(I))--2/ + (L-LPOSI(L1)))/6. END IF END IF 3205 CONTINUE END IF IF(LTN(L1).EC.61THEN DD 3206 I 8 1.NI PP(11 . LMAGA1L1)/BH 3206 CONTINUE END IF 3530 3540 3550 3560 3570 3580 3590 3600 3610 3620 3630 3640 3650 3660 3670 3680 3690 3700 3710 3720 3730 3740 3750 3760 3770 3780 3790 3800 3810 3820 3830 3840 3850 3860 3870 3880 3890 3910 3920 3930 3940 3950 3960 3970 3980 3990 4010 4020 4030 4040 4050 4060 4070 4080 4090 4100 4110 4120 4130 4140 4150 4160 4170 418C 4190 4200 4210 4220 4230 4240 4250 O¢1FIOCTO¢1C)O n 2229 C C --------- PRINT OUT SENDING MOMENT AND AXIAL PRESSURE FOR EVERY LOADING- PRINT 2201.L1 PRINT 2202 DD 3207 =1.NI PRINT 2203.X(I).MM(I).PP(I) 3207 CONTINUE PRINT 2001 C C --------- COMPUTE TOTAL BENDING MOMENT AND AXIAL PRESSURE--- C 00 3208 I=1.NI MIIIIM(II+MM‘I) PIIIsP(I)+PP(I) 3208 CONTINUE 3201 CONTINUE C c --------- PRINT TOTAL BENDING MOMENT AND AXIAL PRESSURE--- C PRINT 2204 PRINT 2202 DO 3209 I-1.NI PRINT 2203. X(I).M(I).P(I) 3209 CONTINUE PRINT 2002 C C --------- OUTPUT FORMAT ----- C 2001 FORMAT(1X.120(“ ~ )/) 2002 FDRMAT(/1X.120('°' )/) 2201 FDRMAT(1X.'MOMENT ANO AXIAL PRESSURE FOR LOADING NUMBER“.IS/) 2202 FDRMAT(1X.'X COORDINATE”.8X.'MOMENT (L8-INCHES)“.2X. Q “AXIAL PRESSURE(PSI)”/) 2203 FDRMAT(1X.F10.5.10X.E20.7. 1X.E20.7) 2204 FORMAT(1X.'MOMENT AND AXIAL PRESSURE FOR TOTAL LOADING“/) KEY- END SU8ROUTINE STRESS(NI.NU.K.8U.KP.SIGMA1.ET.Y2.Y.BL.M.P) K ------ TIME INCREMENT NUMBER. KK ----- ITERATION NUMBER. SIGMA1-STRESSES AT ITERATION K. SIGMA2-STRESSES AT ITERATION K41. EO TTTTT STRAIN AT REFERENCE AXIS. OTHERS'SAME AS MAIN PROGRAM. INTEGER NODEPT(51) REAL DELTAEC(51).E(51).Y(51) REAL SIGMA1(51,51).SIGMA2(51.51).M(51).P(s1),Eo(51),ET(51.51) REAL Y2(51).EC(51.51).NC.NT.LAYERM.MOIFF REAL DEC1(51).X(51) COMMON COMMON COMMON COMMON COMMON COMMON COMMON /8LOCK2/ DELTAT.E.SIGMACC(10).SIGMACT(10).NC(10). NT(10).8C(10).8T(10).ECDOTC(10).ECDOTT(1o) /8LOCK4/ TIME /BLOCK6/ X /BLDCK8/ NPRINT.NODEPT /8LOCK9/ EC /BLOCK10/ ECOMP(10).ETENS(10) /BLOCK7/ ICLAU.MTL(51) 4260 4270 4280 4290 4300 4310 4320 4330 4340 4350 4360 4370 4380 4390 4400 4410 4420 4430 4440 4450 4460 4470 4460 4490 4500 4510 4520 4530 4540 4550 4560 4570 4580 4590 4600 4610 4620 4630 4640 4650 4660 4670 4680 4690 4700 4710 4720 4730 4740 4750 4760 4770 4780 4790 4800 4810 4820 4830 4840 4850 4860 4870 4880 4890 230 DD 3001 I = 1.NI 4900 IF(I.GT.1) THEN 4910 DO 3011 I1-1.I-1 4920 IF(ABS(A8$(M(I))-ABS(M(I1)))/ABS(M(I)).LE.0.0001) THEN 4930 DD 3012 d=1.Nd 4940 ET(I.J)=ET(I1.d) 4950 EC(I.d)*EC(I1.d) 4950 SIGMA1(I.U)=SIGMA1(I1.U) 4970 3012 CONTINUE 4980 12111=v21111 4990 501112501111 5000 GO TO 3001 5010 END I: 5020 3011 CONTINUE 5030 END IF 5040 C 5050 DTNEw-DELTAT 5060 DTPASTs 0. 5070 TIsTIME-DTNEN 5080 C 5090 KM' O 5100 3330 KKcD 5110 3311 KKcKK+1 5120 C 5130 IF(KK.GT.20)THEN 5140 PRINT 2301 5150 GO TO 3000 5160 END IF 5170 C 5180 C --------- COMPUTE DELTA EPSILON C ------- 5190 C 5200 IF(ICLAw.E0.01 THEN 5210 CALL EPSILOA(I.K.NU.TI.DTNEM.OELTAEC.KM.SIGMA1) 5220 ELSE IF(ICLAN.EO.1) THEN 5230 CALL EPSILOEII.K.NJ.TI.DTNEM.OELTAEC.KM.SIGMA11 5240 ELSE 5250 CALL EPSILOC(I.K,NU.TI.DTNEM.DELTAEC.KM.SIGMA1) 5260 END IF 5270 C 5280 C -------- CHECK AND AOUUST TIME INCREMENT IF NEEDED ----- 5290 C 5300 IF(K.GT.O) THEN 5310 IF(KK.EO.1) THEN 5320 RATIO1=A8S